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\title   {
Asymptotically one-dimensional diffusions on scale-irregular gaskets.
         }
\author  {
          Tetsuya Hattori 
  \\ {\small 
          Faculty of Engineering, Utsunomiya University, 
} \\ {\small 
          Ishii-cho, Utsunomiya 321, Japan
} \\ {\small e-mail: 
          hattori@tansei.cc.u-tokyo.ac.jp
}        }
\begin{document}
\maketitle
\begin{abstract}
A simple class of fractals which lack exact self-similarity is 
introduced, and the asymptotically one-dimensional diffusion process is 
constructed.
The process moves mostly horizontally for very small scales, while for large 
scales it diffuses almost isotropically, in the sense of the off-horizontal 
relative jump rate for the decimated random walks of the process.
\par
An essential step in the construction of diffusion 
is to prove the existence of appropriate time-scaling factors.
For this purpose, a limit theorem for a discrete-time multi-type 
supercritical branching processes with singular and irregular 
(varying) environment, is developed.
%
\bigskip \par \noindent {\em Mathematics Subject Classification: } 
60J60, 60J25, 60J85, 60J15
\end{abstract}
\bigskip \par \noindent {\bf Running title: 
Asymptotically one-dimensional diffusion on scale-irregular gasket}
 
%----------------------------------------------------------------------
\section{Introduction.} \seca{s1}
% rnd1 (v940529)
\fgl{abb}
In this paper we construct 
asymptotically one-dimensional diffusion processes 
on the \rgs.
We generalize the \sg, a triangle based fractal, to define the \rgs, 
which we introduce as simple examples of
finitely ramified fractals that are scale-irregular, 
i.e. do not have exact self-similarity.
Let $\{ (a_N,b_N),\ N\in\integers \}$ be
a sequence of pair of positive integers.
Let $\Fgasket'_0$ be the set of vertices and edges of a triangle, 
considered as a graph.
Join $a_0+2b_0\ \Fgasket'_0$s' as in \figu{abb} to form a triangle 
$\Fgasket'_{-1}$, so that one side of $\Fgasket'_{-1}$ is composed of 
$a_0+1\ \Fgasket'_0$s' and the 
other two sides are composed of $b_0+1\ \Fgasket'_0$s'.
Repeat this procedure recursively to obtain $\Fgasket'_{-n-1}$ from 
$a_{-n}+2b_{-n}\ \Fgasket'_{-n}$s', for $n\in\pintegers$.
We regard one of the three vertices of $\Fgasket'_{-n}$ to be the origin $O$
and regard $\Fgasket'_{-n} \subset \Fgasket'_{-n-1}$.
Let $\Fgasket_0$ be the two copies of the infinite graph 
(extending infinitely outward)
${\displaystyle \bigcup_{n=0}^{\infty} \Fgasket'_{-n}}$
joined at the origin $O$.
Next, define $\Fgasket_1 \supset \Fgasket_0$ by adding a structure 
(bonds and vertices) 
similar to the structure of $\Fgasket'_{-1}$ in \figu{abb}, but with 
$(a_0,b_0)$ replaced by $(a_1,b_1)$,
to each of the unit triangle.
Repeat this procedure recursively inwards and add
substructure of $a_{N+1}+2b_{N+1}$ triangles to each of the smallest 
triangle in $\Fgasket_N$ to obtain $\Fgasket_{N+1} \supset \Fgasket_N$.
We call the graph $\Fgasket_N$ a pre-gasket (of scale $N$).
Let $G_N$ denote the set of vertices in $\Fgasket_N$.
Each element of $G_N$ (including $O$) has
four $N$-neighbor points; by an $N$-neighbor point we mean a vertex 
in $\Fgasket_N$ joined by an edge of $\Fgasket_N$.
If $N>N'$ then $G_N \supset G_{N'}$.
Put $\dsp \lambda_n=\prod_{k=1}^{n} \min\{a_k+1,b_k+1\}^{-1}$, 
$n\in\pintegers$.
We introduce metric $d$ such that $d(x,y)=\lambda_n$ if $x$ and $y$ are 
$n$-neighbor points.  
See \appu{sa}.
$\dsp G=\overline{\bigcup_N G_N}=\overline{\bigcup_N \Fgasket_N}$ is the \rg.
The definitions are independent of embedding into Euclidean spaces, but
for simplicity, 
the terminology for subsequent explanations will be
as if a \rg\ $G$ is embedded in $\twreals$ such that 
$x$-axis is contained in $G$ and $G$ is contained in upper half plane
(e.g. horizontal means parallel to $x$-axis).
If $a_N$ and $b_N$ are independent of $N$,
the corresponding fractal has exact self-similarity.
The \sg\ is the \rg\ with $a_N=b_N=1,\ N\in\integers$.
If $b_N=a_N$,  $N\in\integers$, which we call the \rags, 
there is an obvious local spatial symmetry (isotropy) shared with \sg.
\par
For a process $X$ taking values in $G$ we define
$T_{n,i}(X)\,,\ n\in\integers\,$, 
(the time that $X$ hits $G_n$ for the $i+1$-th time, 
counting only once if it hits the same point more than once in a row)
by 
$T_{n,0}(X)= \inf \left\{ t \ge 0 \, |\ X(t) \in G_n \right\}\,$, and
\[ T_{n,i+1}(X)= \inf \left\{ t > T_{n,i}(X) \, |\ 
X(t) \in G_n \setminus \left\{ X(T_{n,i}(X)) \right\} \right\} \,,\ 
i=0,1,2,\cdots. \]
For an integer $n$ and a Markov process $X$ on $G$ or on $G_N$ 
for some $N\ge n$,
we call a simple random walk $X^{(n)}$ on $G_n$ defined by 
$X^{(n)}(i) = X(T_{n,i}(X))$ the $n$-decimated walk of $X$.
By definition, 
\prpb\prpa{consistency}
If $n<N$ and 
$X^{(n)}$ and $X^{(N)}$ are the $n$ and $N$-decimated walks of $X$ 
respectively, then $X^{(n)}$ is the $n$-decimated walk of $X^{(N)}$.
\prpe
 
For $N\in\integers$ and $w>0$, we define a simple random walk 
$X_{N,w}$ on $G_N$ as follows.
At each integer time, the random walker 
jumps to one of the four $N$-neighbor points,
and the relative rates of the jumps are
$1$ (resp. $w$) for a jump in horizontal 
(resp. $\pm 60^{\circ}$ or $\pm 120^{\circ}$) direction.
We prove in this paper the following.
\thmb \thma{main}
Assume that $\{ (a_N,b_N),\ N\in\integers \}$ 
is a bounded sequence of pairs of integers satisfying 
\eqnb \eqna{simple}
a_N \ge 2,\ b_N \ge 2,\ b_N<2 a_N,\ \ N\in\integers,
\eqne
and let $G$ be the \rg\ defined by this sequence.
Then there exist a sequence of positive numbers $w_N$, $N\in\integers$
satisfying 
\eqnb \eqna{a1}
\limf{N} w_N = 0 ,
\eqne
and a symmetric Feller process $X$ 
(a continuous $G$-valued strong Markov process whose transition operators
map the space of bounded continuous functions into itself,
symmetric with respect to a measure $\mu$ on $G$ defined by
$\int f \,d\mu = {\limf{N}} \left(\prod_{k=0}^{N} (a_k+2b_k)^{-1}\right)
 \sum_{x\in G_N} f(x)$),
such that 
for $N\in\integers$,
the $N$-decimated walk of $X$ is equal in law to the random walk $X_{N,w_N}$.
\thme
 
The assumptions \eqnu{simple} are to avoid complications.
We will prove \thmu{main} for any $w_0$ satisfying
\eqnb \eqna{asymp}
w_0 \in I \defd (0,\inf_N \{2a_N/b_N\} -1).
\eqne
 From \prpu{consistency} and \thmu{main},
the $(N-1)$-decimated walk of $X_{N,w_N}$ is equal in law to $X_{N-1,w_{N-1}}$,
 from which it follows that the sequence $\{w_N\}$ satisfies a recursion
relation
\eqnb \eqna{wrec}
w_{N-1}=f_{(a_N,b_N)}(w_N)\,,\ N\in\integers,
\eqne
where
\eqnb \eqna{frec}
f_{(a,b)}(w)=\frac{2w\{(1+a)b+(a\,b+a+b)w\}}{b(b+2)+2(b^2+a+b)w+b^2 w^2}\,.
\eqne
Proof of \eqnu{wrec} is elementary (but lengthy), and is similar
to that of \cite[Proposition 1.1]{HHW93}.
It is elementary to see that
\prpb \prpa{wasymp}
If \eqnu{simple} and \eqnu{asymp} are satisfied, 
there exists one and only one sequence $\{w_N\}$ which satisfies \eqnu{wrec}
and which is in the open interval $I$.
Moreover, 
$\{w_N\}$ is strictly decreasing and satisfies \eqnu{a1}.
$ \dsp \limf{n} \frac{w_{n+s}}{w_n} \prod_{k=1}^{s} \delta_{n+k} =1$
uniformly in $s\in\pintegers$, where
$\dsp \delta_k \defd \frac{2(1+a_k)}{2+b_k} >1$.
For \rags\ (i.e. $b_N=a_N$), it also holds that $I=(0,1)$ and
$\dsp \lim_{N\to -\infty} w_N=1$.
\prpe
 
The ratio of the rate for a off-horizontal to horizontal jump of $X_{N,w_N}$ 
is $w_N$, hence 
\eqnu{a1} means that on small scales the process favors 
horizontal moves, while $w_N>0$ means that the process span the
whole fractal space and is not confined in a line.
For spatially symmetric \rags, $\lim_{N\to -\infty} w_N=1$,
which implies that isotropy is asymptotically restored.
Not much studies has been done on this diffusion.
The asymptotically one-dimensional diffusion is first constructed
on the \sg\ \cite{HHW93}.
\cite{V} gives a characterization of the 
diffusion on the \sg\ by exit distributions.
\par
The fractals may be regarded to have ^^ obstacles' or holes in the space,
when compared to uniform Euclidean spaces.
Intuitively, 
a random walker that favors horizontal motion 
performs a one-dimensional random walk between
a pair of obstacles, and eventually is forced to move in 
off-horizontal direction before they could move further horizontally.
There are obstacles of various scales (sizes), separated by distances of
the same order as their scales, hence globally, the random walker
is scattered almost isotropically.
This picture may be regarded as a new mechanism of 
homogenization, unique to fractals.
The intuitive picture of isotropy restoration
can be made mathematically persuasive
through analyses of renormalization groups \cite{HHW93,HHW87}.
The intuition implicitly guided the studies in \cite{HHW93}, but
in spite of the generality in the intuition,
it was not clear how to obtain such
diffusions for fractals which lack exact self-similarity.
Also the statements for the diffusion in that work were not referring to the 
properties which explicitly embodied the picture.
It is the purpose of this paper to report some positive answers (\thmu{main})
to these points.
\medskip\par
We construct the diffusion as a weak limit of 
$X_{N,w_N}([L_N t])$, $N\in\integers$, 
for a time-scaling constant $L_N$.
A key step in the construction is an asymptotic estimate
of number of steps of $X_{N,w_N}$, whose expectation value is $L_N$.
We apply a limit theorem for discrete-time multi-type supercritical 
branching processes with singular and irregular environment.
We need multi-type branching processes because the horizontal and 
off-horizontal jumps have different transition probabilities.
Branching rates change with the generation $N$ (irregular environment)
because the substructure of a triangle in the pre-gasket varies with 
its scale $N$.
Environment varies also because the transition probabilities
of the random walk $X_{N,w_N}$ vary with $N$.
In particular, birth rates of types corresponding to
the numbers of off-horizontal jumps approach $0$ as $N\to\infty$ 
(singular environment).
Compared with existing related results, 
there are two major complications arising from these requirements;
criterion for supercriticality, and 
scaling factor for total number of descendants.
For a construction of spatially symmetric diffusion on an
exact self-similar finitely ramified fractal \cite{Kus,BP,L,Kum}, the 
expectation of off-spring for the associated branching process
is a constant matrix independent of generation $N$.
For a construction of asymptotically one-dimensional diffusion on
an exact self-similar finitely ramified fractal \cite{HHW93,HW}, 
the off-spring expectation matrix has a limit as $N\to\infty$.
In these cases,
the largest eigenvalue of the (limit of) off-spring expectation matrix 
gives the (asymptotic) growth rate
of descendant expectations and governs supercriticality.
A pioneering work for scale-irregular fractals
by Hambly \cite{Ham} deals with 
spatially symmetric diffusions on fractals called $HSG(\bar{\nu})$ 
(which have much in common with \rags\ as far as construction of 
diffusions are concerned).
Due to the spatial symmetry,
the associated branching process is of one-type, hence
the off-spring expectation is one-dimensional, which gives the growth rate.
In the present study,
off-spring expectation is a multi-dimensional matrix, and 
neither is constant nor has a limit.
Thus a criterion for supercriticality cannot be given in terms of 
growth rates.
Furthermore, the ratios 
of expectations of the population between different types are unbounded,
which obscures at first site, the existence of 
scaling factor for total descendant numbers.
\par
Our approach is partly inspired by a study on multi-type branching processes 
in random environment by Cohn. 
Much of our proof of \prpu{Cohn} follows the idea in \cite{Cohn}.
In that work, a probability measure on environments is considered, and
the assumptions on stationarity and ergodicity
implicitly assured the last two assumptions in \prpu{Cohn} 
(including supercriticality) to hold almost surely.  
This approach is not suitable for our purpose to consider singular 
environment, where some of the branching rates vanish in the limit.
To formulate a sufficient condition of supercriticality in \thmu{limit},
we introduce a recursion relation in \appu{sd} which reflects
recursive nature of branching processes.
We apply this recursion relation also to prove 
continuity of limit distribution in \thmu{cont}.
The idea of using recursion relation 
to prove continuity originally appeared in \cite[Lemma 2.7]{HHW93},
which we refine to put it into a framework of branching processes, 
and generalize to handle irregular environments.
It turns out in \prpu{total} that
the existence of scaling factor for total descendant
follows from a fact that the distribution of normalized population
converge to a limit independent of types.
Consideration on the branching processes may be interesting in its 
own respect, so we will discuss this in \secu{s3} independently 
of other sections.
\par
To apply the general theory of branching processes to the diffusion, we 
consider in \secu{s2} estimates for generating functions.
The calculations are lengthy,
for some of which the author used REDUCE, 
a computer program for rigorous formula reduction (\prpu{REDUCE}).
We use these basic estimates to obtain 
estimates for number of steps of $X_{N,w_N}$,
to which one can apply \cite[Sect. 3]{HHW93}.
 
%----------------------------------------------------------------------
\section{Branching process with singular and irregular environment.} \seca{s3}
% rnd3 (v940529)
 
Let $d\ge 2$ be an integer and put $\edges\defd \{1,2,3,\cdots,d\}$.
Consider a discrete time
$d$-type branching process $\vZ_N = (Z_{N,j}$, $j\in\edges)$, 
$N\in\pintegers$.
Given $n\in\pintegers$ and $i\in\edges$, let 
\[ \vZ_{n,N,i} \defd ( Z_{n,N,i,j}\,,\ \ j\in\edges),\ \ \ N=n,n+1,\cdots,\]
be random vectors which give the distribution of 
the number of descendant at time $N$ 
 from a single ancestor of type $i$ at time $n$.
We have, for $r\in\pintegers$ and $j\in\edges$,
\[ Z_{n+r,j}= \sum_{i\in\edges} \sum_{u=1}^{Z_{n,i}} Z_{n,n+r,i,j,u} \,, \]
where $(Z_{n,n+r,i,j,u}\,,\ j\in\edges)$, $u\in\pintegers$, are 
i.i.d.\ copies of $\vZ_{n,n+r,i}$ when conditioned on $Z_{n,i}$.
Let $\{e_n\}$ be a non-negative sequence.
\prpb \prpa{Cohn}
Assume that following three conditions hold for each $i\in\edges$.
\itmb
\item
Uniform estimates for second moments of
$\dsp W_{nNij} \defd \frac{Z_{nNij}}{\EE{Z_{nNij}}}$;
\[ \arrb{c} \AHFA \dsp
v \defd \sup_{n\in\pintegers} \sup_{N\ge n+n_0} e_n\,\EE{W_{nNij}^2} 
< \infty,\ \ j\in\edges\,, 
\\ \AHFA \dsp
\limf{p} \sup_{N\ge n+n_0} \EE{W_{nNij}^2;\ W_{nNij}>p} =0,\ \ 
j\in\edges, n\in\pintegers\,,
\arre \]
for some constant $n_0\in\pintegers$.
\item
For each $n\in\pintegers$,
$\dsp \gamma_{ni} \defd \limf{N} \frac{\EE{Z_{nNij}}}{\EE{Z_{nN1j}}} > 0$
exist, positive and independent of $j\in\edges$. 
\item
$e_N \,Z_{Nj}$ diverges in probability to infinity; 
$\dsp \lim_{N\to\infty} \prb{e_N \,Z_{Nj} \ge p} = 1$, $j\in\edges$, $p>0$.
\itme
Then the sequence of normalized random vectors
$(Z_{Nj}/\EE{Z_{Nj}},\ j\in\edges)$ converges in $L_2$ as $N\to\infty$
to a random vector $(W,W,\cdots,W)$ with $\EE{W}=1$.
\prpe
Proofs of the results in this section is postponed to
the end of the section.
Generalization to include $e_N$ is for our application in \secu{s2}.
A simple sufficient condition for the existence of $\gamma_{ni}$
is given in \appu{sc},
in terms of off-spring expectation matrices $A_N$ 
($ A_{Nij} = \EE{Z_{N-1,N,i,j}}$).
The last assumption states supercriticality.
One of our main concern here is a useful condition for 
the last assumption to hold.
\defb
A family of sequences of pairs of reals
$\{(x_{k,n},y_{k,n})$, $n=0,1,\cdots,k\}$, $k\in\pintegers$,
is said to satisfy the assumption R, if there exist 
sequences of non-negative numbers 
$\{a_n\}$, $\{w_n\}$, $\{w'_n\}$, $n\in\pintegers$, satisfying 
$ 2 \le \inf_n a_n$, $\sup_n a_n < \infty$, and
$\dsp \max\{w_n,w'_n\} \le \min\{1,\ C_w \delta^{-n}\}$, $n\in\pintegers $,
for constants $C_w>0$ and $\delta > 1$, 
such that
\eqsb
x_{k,n} &\le& x_{k,n+1}^{a_{n+1}}+w_{n+1} \, 
\min\{1-x_{k,n+1}^{a_{n+1}}\,,\ y_{k,n+1}\} \,,\ \ \\
y_{k,n} &\le& x_{k,n+1}+ w'_{n+1} \, y_{k,n+1} \,,\ \ \ \ 0\le n\le k\,,
\eqse
hold for all $k\in\pintegers$.
Similarly, 
$\{(x_{k,n},y_{k,n})$, $n\in\pintegers\}$, $k\in\pintegers$,
is said to satisfy the assumption R,
if similar relation hold for $n\in\pintegers$ and $k\in\pintegers$.
\defe

\thmb \thma{limit}
Assume that for some $j_0\in\edges$,
$Z_{0,j}=1$, $j=j_0$, and $Z_{0,j}=0$, otherwise.
Let $p>0$ and $j\in\edges$.
Suppose that there exists an integer $n_0$ and 
a non-empty subset $\edges' \subset \edges$, 
not equal to $\edges$,
such that the family of sequences $\{(x_{k,n},y_{k,n})\}$ defined by 
\eqsb
x_{k,n} &=& \max_{i\in\edges'} \prb{e_{n_0+k} \,Z_{n,n_0+k,i,j}<p} \,,
\\
y_{k,n} &=& \max_{i\in\edges\setminus\edges'} 
\prb{e_{n_0+k} \,Z_{n,n_0+k,i,j}<p} \,,\ \ \ 0\le n\le k,\ k\in\pintegers\,,
\eqse
satisfies the assumption R, and
\eqnb \eqna{Rxkk}
\liminf_{k\to\infty,\ x_{k,k}\ne 0} 
\left\{ (-\log x_{k,k}) \prod_{\mu=0}^{k} a_{\mu}\right\}^{1/k} >1 \,. 
\eqne
Then 
\[ \lim_{N\to\infty} \prb{e_{N} \,Z_{N,j} \ge p} = 1 \,. \]
\thme
The assumption \eqnu{Rxkk} is an ^^ a priori estimate' that
$\prb{e_{n_0+k} Z_{k,n_0+k,i,j}<p}$ is not too large.
The Theorem then says that it is in fact small.
Let us call the types $j\in\edges'$ the dominant types,
and the types $j\not\in\edges'$ the recessive types.
The assumption R reflects the recursive nature of branching process.
It is satisfied when 
the probability that recessive types appear in the
off-springs of a parent at generation $n$ 
vanishes exponentially as $n\to\infty$,
and if recessive type do not appear in the off-springs, then 
at least one dominant type off-spring appears from a recessive parent,
while at least $a_{n+1}$ (no less than $2$ but bounded) dominant off-springs
appear from a dominant parent.
$\{w_n\}$ represents singular environment,
while $n$-dependence of $\{a_n\}$ implies irregular environment.
Though the branching rate to recessive types vanish in the limit, 
the recessive types may contribute significantly to the growth of
dominant types, because 
a recessive parent may give birth to exponentially many dominant type
off-springs.  Thus in general, we can not discard the recessive types
 from consideration for limit theorems.
Note also that supercriticality is non-trivial because of the recessive types.
Assumption on initial condition $Z_{0,j}$ is chosen to be simple,
to avoid complications.
\par
The following is useful in obtaining 
an a priori estimate of type \eqnu{Rxkk} from moments of $Z_{nNij}$.
\prpb \prpa{Schwarz}
Let $p\in\reals$ and $X$ a real valued random variable.
If $\EE{X} > p$ then $\dsp \prb{X \le p} \le 1-d+\sqrt{d^2-1}$,
where $d=1+2^{-1} \,\VV{X} \, (\EE{X}-p)^{-2}$.
\prpe
 
The next statement is on
the existence of norming factor for total descendant numbers.
\prpb \prpa{total}
Assume that the sequence 
$\dsp (Z_{N,j}/\EE{Z_{N,j}}$, $j\in\edges)$, $N\in\pintegers$,
converges in probability as $N\to\infty$
to a random vector $(W,W,\cdots,W)$.
Then 
$\dsp \frac{\sum_{j\in\edges} Z_{N,j}}{\sum_{j\in\edges} \EE{Z_{N,j}}} $
converges in probability as $N\to\infty$ to $W$.
\prpe
 
We complete our list of the results with a sufficient condition for
the continuity of the limit distribution,
stated in terms of the assumption R.
Let $W_{n,i}$, $n\in\pintegers$, $i\in\edges$, be real valued 
random vectors, and let
\[ \Phi_{n,i}(t) \defd \EE{\exp \left( \sqrt{-1} \, t \, W_{n,i}\right)}\,, \]
denote the characteristic function.
We assume an ^^ a priori' estimate of the form
\eqnb \eqna{apriori}
|\Phi_{n,i}(t)| \le 
1 - C_{n} \, t^2 \,,\ \ -t_n < t < t_n\,,\ n\ge n_0\,,\ i\in\edges',
\eqne
for some non-empty subset $\edges' \subset \edges$ not equal to $\edges$, 
an integer $n_0$, and positive reals $C_{n}$ and $t_n$.
\thmb \thma{cont}
Assume that $\{t_k,\ k\in\pintegers\}$ in \eqnu{apriori}
diverges to infinity as $k\to\infty$ exponentially fast at most
($\dsp \limsup_{k\to\infty} t_k^{1/k} < \infty$), and satisfies 
\[ \theta \defd \inf_{k;\;\exists t_j<t_k} \sup_{j;\;t_j<t_k} 
\frac{t_j}{2 \, t_k} >0 \,. \]
If for any sequence of reals $\{s_k,\ k\in\pintegers\}$ 
the family of sequences $\{(\tilde{x}_{k,n},\tilde{y}_{k,n})$, 
$n\in\pintegers\}$, $k\in\pintegers$, defined by 
\eqsb
\tilde{x}_{k,n} &=& \max_{i\in\edges'} \left| \Phi_{n_0+n,i}(s_k) \right|
\\
\tilde{y}_{k,n} &=& \max_{i\in\edges\setminus\edges'}  
\left| \Phi_{n_0+n,i}(s_k) \right| \,,
\ \ \ n\in\pintegers\,,\ k\in\pintegers\,,
\eqse
satisfies the assumption R, and if
\[ \liminf_{k\to\infty} 
\left( t_k^2 \, C_{k} \, \prod_{\mu=0}^{k-n_0} a_\mu \right)^{1/k} > 1 \,, \]
holds with $a_{\mu}$ as in the assumption R,
then the distribution of $W_{n,i}$ is continuous for all $n\in\pintegers$
and $i\in\edges$.
\thme
The intuition for the assumptions is similar to those for \thmu{limit},
with $W_{n,i}$ being a weak limit of 
$\dsp \frac{\sum_j Z_{nNij}}{\sum_{k,j} \EE{Z_{0Nkj}}}$.
\smallskip\par
The rest of this section is devoted to the proofs of the stated results.
\prfofb{\prpu{Cohn}}
Since $W_{nNij} \ge 0$ and $\EE{W_{nNij}}=1$, the family of random variables
$\dsp \{W_{nNij}\,,\ N=n,n+1,\cdots\}$ is tight, hence there exists a
subsequence of integers $\{k_N\}$ such that $W_{n,k_N,i,j}$ converges 
weakly as $N\to\infty$ to a random variable $\WW_{nij}$.
By assumption $\EE{W_{nNij}^2}$ is uniformly bounded in $N$, which,
with the tightness, implies that $\{W_{nNij}\}$ is uniformly integrable.  
The assumption also implies that $\{W_{nNij}^2\}$ is uniformly integrable.
Weak convergence and uniform integrability imply convergence of 
expectations;
\eqnb \eqna{expec}
\EE{\WW_{n,i,j}} = \limf{N} \EE{W_{n,k_N,i,j}} =1\,.
\eqne
\eqnb \eqna{varia}
\sup_{n\in\pintegers} e_n\,\EE{\WW_{n,i,j}^2} \le v \,.
\eqne
Put 
$\dsp W_{n,N,i,j,u} \defd \frac{Z_{n,N,i,j,u}}{\EE{Z_{n,N,i,j,u}}}$.
$(W_{n,N,i,j,u}\,,\ j\in\edges)$, $u\in\pintegers$, are 
i.i.d.\ copies of $(W_{n,N,i,j}\,,\ j\in\edges)$ when conditioned on $Z_{n,i}$.
Hence 
$(W_{n,k_N,i,j,u}$, $j\in\edges$, $u\in\pintegers)$, 
conditioned on $Z_{n,i}$, converges weakly
as $N\to\infty$ to a random vector 
$(\WW_{n,i,j,u}$, $j\in\edges$, $u\in\pintegers)$,
where 
$(\WW_{n,i,j,u}\,,\ j\in\edges)$, $u\in\pintegers$, are 
i.i.d.\ copies of 
$(\WW_{n,i,j}\,,\ j\in\edges)$ when conditioned on $Z_{n,i}$.
Hence \eqnu{expec} and \eqnu{varia} imply, for positive integer $p$,
\[\arrb{l} \AHFA \dsp
\EE{\left( Z_{ni}^{-1}\,\sum_{u=1}^{Z_{ni}} (\WW_{niju}-1) \right)^2;\ 
Z_{ni}\ge p}
= \sum_{q=p}^{\infty} q^{-1} \EE{(\WW_{nij1}-1)^2\,;\ Z_{ni}=q}
\\ \le (v/e_n+1)/p\,.
\arre\]
This implies, with
the assumption on supercriticality and Chebyshev's inequality,
that 
for $\eta>0$ and $\epsilon>0$,
there exists $n_0$ such that if $n\ge n_0$ then
\[\arrb{l} \AHFA \dsp
\prb{\left| Z_{ni}^{-1}\,\sum_{u=1}^{Z_{ni}} (\WW_{niju}-1) \right|
\ge \epsilon}
\\ \AHFA \dsp
\le \prb{\left| Z_{ni}^{-1}\,\sum_{u=1}^{Z_{ni}} (\WW_{niju}-1) \right|
\ge \epsilon\,,\ Z_{ni}\ge \frac{2(v+e_n)}{\eta\,\epsilon^2\,e_n}}+\eta/2
\ \le\ \eta \,.
\arre\]
Convergence in probability follows;
\eqnb \eqna{inprob}
\limf{n} \prb{\left| Z_{ni}^{-1}\,\sum_{u=1}^{Z_{ni}} \WW_{niju}-1 \right|
\ge \epsilon} = 0 \,,\ \ \epsilon>0.
\eqne
 
The second assumption in the statement and a property of branching process
\eqnb \eqna{brprexpec}
\EE{Z_{N,j}}= \sum_{k\in\edges} \EE{Z_{n,k}} \, \EE{Z_{n,N,k,j}} 
\eqne
implies that the limit
\[ \beta_{n,i} \defd \limf{N} \frac{\EE{Z_{n,k_N,i,j}}}{\EE{Z_{k_N,j}}}
=\frac{\gamma_{n,i}}{\sum_{k} \EE{Z_{n,k}} \, \gamma_{n,k}} \,, \]
exists, positive, independent of $j$, and satisfies
$\dsp \sum_{i\in\edges} \beta_{ni} \, \EE{Z_{ni}} = 1 $.
This, with \eqnu{inprob} and the non-negativity of $Z$ and $\WW$ implies
convergence in probability,
\eqnb  \eqna{bWZ}
\limf{n} \prb{ \left|
\sum_{i\in\edges} \left(\beta_{ni} \, \sum_{u=1}^{Z_{ni}} \WW_{niju} \right) 
- \sum_{i\in\edges} \beta_{ni} \, Z_{ni} 
\right| > \epsilon }=0 \,,
\eqne
for $\epsilon>0$.
Note that in \eqnu{bWZ} everything except possibly $\WW_{niju}$
is independent of the choice of subsequence $\{k_N\}$.

Put
\[ \xi_n(x) \defd \limf{N} \prb{ Z_{k_N,j}/\EE{Z_{k_N,j}} 
\le x_j\,,\ j\in\edges \mid \vZ_n} \,. \]
$\xi_n(x)$ is a bounded martingale, hence
converges as $n\to\infty$ to a random vector $\xi(x)$ a.s.
The definitions of $\WW_{niju}$ and $\beta_{ni}$ with \eqnu{brprexpec}
imply 
\[ \xi_n(x) = \prb{ 
\sum_{i\in\edges} \left(\beta_{ni} \, \sum_{u=1}^{Z_{ni}} \WW_{niju} \right) 
\le x_j\,,\ j\in\edges \mid \vZ_n} \,, \]
on set of continuity points.
This with \eqnu{bWZ} implies that $\xi(x)$ is independent of the choice
of subsequence $\{k_N\}$.
In particular, 
\[ \EE{\xi(x)}= \limf{N} \prb{Z_{k_N,j}/\EE{Z_{k_N,j}} \le x_j\,,\ 
j\in\edges } \]
is independent of the subsequence, hence
$Z_{N,j}/\EE{Z_{N,j}}$ converges weakly to a random vector with distribution
function $\EE{\xi(x)}$. 
Furthermore, \eqnu{bWZ} implies that this random vector has
equal components.
Convergence in probability, and then in $L_2$, is now proved 
exactly as in \cite[step 4]{Cohn}.
\prfofe
 
\prfofb{\thmu{limit}}
By definition $0\le x_{k,n} \le 1$ and $0 \le y_{k,n} \le 1$ for all 
$n$ and $k$.  
With the assumption R and \eqnu{Rxkk}, 
we see that $\{(x_{k,n},y_{k,n})\}$ satisfies
all the assumption of \thmu{recineq}.
\thmu{recineq} implies $\dsp \lim_{k\to\infty} \max\{x_{k,0},\ y_{k,0}\}=0$,
which gives
$\dsp \lim_{N\to\infty} \prb{e_N Z_{0Nij} \ge p} = 1$.
\prfofe
 
\prfofb{\prpu{Schwarz}}
Put $Y=X-\EE{X}$, $v=\VV{X}=\VV{Y}$, $b=\EE{X}-p>0$, and 
$t=\prb{Y>-b}=\prb{X>p}$.
$\dsp 0=\EE{Y} \le \EE{Y;\ Y\le -b} + \EE{Y;\ Y>0} $
implies $\dsp \EE{Y;\ Y>0} \ge b \, (1-t)$.
Using Schwarz inequality we have
\[t\,v \ge \prb{Y>0}\,\EE{Y^2;\ Y>0} \ge (\EE{Y;\ Y>0})^2 \ge b^2\,(1-t)^2\,.\]
The statement follows by solving this algebraic inequality in $t$.
\prfofe
 
\prfofb{\prpu{total}}
\eqsb \lefteqn{
\limf{N} \EE{\min\{1,\ \left|
\frac{\sum_{j\in\edges} Z_{N,j}}{\sum_{j\in\edges} \EE{Z_{N,j}}}-W \right| \}}
} \\ &\le& \limf{N} \EE{\sum_{j} \frac{\EE{Z_{N,j}}}{\sum_k \EE{Z_{N,k}}} 
\min\{1,\ \left| \frac{Z_{N,j}}{\EE{Z_{N,j}}} -W \right| \} }
\\ &\le& \sum_{j\in\edges} \limf{N} \EE{\min\{1,\ 
\left| \frac{Z_{N,j}}{\EE{Z_{N,j}}} -W \right| \} } = 0 \,.
\eqse
The assumption implies the last equality.  Hence the statement follows.
\prfofe
 
\prfofb{\thmu{cont}}
Note that the definition and assumption on $\theta$ imply $0< \theta < 1/2$.
Let $n_1 \ge n_0$, and let $\{s_k,\ k\in\pintegers\}$ be a sequence of reals 
satisfying
\eqnb \eqna{tst}
\theta \, t_{n_1+k} \le |s_k| \le t_{n_1+k}\,,\ \ k\in\pintegers\,. 
\eqne
Put
\eqsb
x_{k,n} &=& \max_{i\in\edges'} \left| \Phi_{n_1+n,i}(s_k) \right|
\\
y_{k,n} &=& \max_{i\in\edges\setminus\edges'}  
\left| \Phi_{n_1+n,i}(s_k) \right| \,,
\ \ \ 0\le n \le k,\ k\in\pintegers\,.
\eqse
The assumption R implies that
$\{(x_{k,n},y_{k,n})\}$
satisfies the recursion relation in the
assumption of \thmu{recineq}.
with $\{w_{n_1+n-n_0}\}$ and $\{a_{n_1+n-n_0}\}$ in place of
$\{w_{n}\}$ and $\{a_{n}\}$.
Also \eqnu{tst} and the assumptions on a priori estimates imply
\[ \liminf_{k\to\infty}
\left\{ -\log x_{k,k} \prod_{\mu=0}^{k} 
a_{n_1+\mu-n_0}\right\}^{\frac{1}{k}} \ge \liminf_{k\to\infty}
\left\{ \theta^2  C_k  t_k^2  \prod_{\mu=n_1-n_0}^{k-n_0} 
a_{\mu}\right\}^{\frac{1}{k-n_1}} >1 . 
\]
We see that $\{(x_{k,n},y_{k,n})\}$ satisfies 
all the assumption of \thmu{recineq}, hence 
there exist positive constants $C_1$ and $C_2$ (which may depend on $n_1$ but
not on $k$) such that
\eqnb \eqna{expdecay} 
\left| \Phi_{n_1,i}(s_k) \right| \le C_1 \, \exp(-C_2 \, k^2)\,,
\ \ \ k\in\pintegers\,,\ i\in\edges\,. 
\eqne
 
Let $t\in\reals$ be a number satisfying $|t|> \min_j \{ t_j \}$,
and let $t_k=\min\{t_j \mid t_j>|t|\}$.
(The assumption $\limf{k} t_k =\infty$ implies that the minimum exists.)
The definition of $\theta$ implies that there exists a $t_j$
satisfying 
$\dsp t_k > t_j > \theta \, t_k $.
The choice of $t_k$ therefore implies that 
\eqnb \eqna{tk}
\theta \, t_k < |t| < t_k \,.
\eqne
The assumption $\dsp \limsup_{k\to\infty} t_k^{1/k} < \infty$
implies that there exist positive constants $C_3$ and $C_4$ such that
$t_k < C_3 \, C_4^k$, $k\in\pintegers$.
With \eqnu{tk} we have $k > (\log |t| - \log C_3)/\log C_4$.
This and \eqnu{tk}, \eqnu{tst}, and \eqnu{expdecay} imply
\eqnb \eqna{expd2}
\left| \Phi_{n_1,i}(t) \right| \le C_5 \, \exp(-C_6 \, (\log |t|)^2)\,,
\eqne
for sufficiently large $|t|$, with positive constants $C_5$ and $C_6$
independent of $t$.
Therefore, if $n\ge n_0$, then $\Phi_{n,i}\in L_1(\reals)$ for 
all $i\in\edges$.
By the assumption of recursion relation, it follows inductively that
$\Phi_{n,i}\in L_1(\reals)$ for any $n \in \pintegers$, which implies that
$W_{n,i}$ is continuous.
\prfofe
 
%----------------------------------------------------------------------
\section{Convergence of path measures.} \seca{s2}
% rnd2 (v940529)
%\\cv1-8
\fgl{type}
 
Consider a pre-gasket $\Fgasket_N$ and its vertices $G_N$.
Modulo translation in $\twreals$ and reflection with respect to 
vertical lines, there are $4$ types of points $A$, $B$, $D$, and $E$ 
in $G_{N}$ and $8$ types of edges (seen as ordered pair of points) 
$Ap$, $Ar$, $Bp$, $Bq$, $Br$, $Dq$, $Ep$, and $Er$, 
in $\Fgasket_{N}$ (\figu{type}).  
We put 
\[ \edges \defd \{ Ap,\ Ar,\ Bp,\ Bq,\ Br,\ Dq,\ Ep,\ Er \}\,. \]
Note that these definitions can be taken independent of $N$.
We will use the same notation without explicitly specifying $N$.
\par
Let $(a,b)$ be a pair of positive integers, and 
consider the case that $\Fgasket_N$-substructure of the pre-gasket 
$\Fgasket_{N-1}$ is parametrized by $(a,b)$:  In the notation of \secu{s1}, 
$(a_N,b_N)=(a,b)$.
Let $\Paths(a,b,i)$ be the set of 
walks on $G_N$ whose starting point $X$ and stopping point form an edge of
type $i\in\edges$ in $\Fgasket_{N-1}$, and such that do not pass
through points in $G_{N-1} \setminus \{X\}$:
\[ \arrb{r} 
\Paths(a,b,i)=\{ \paths=(\paths(0),\cdots,
\paths(L)) \subset G_N \ \mbox{for some $L$} \mid 
(\paths(0),\paths(L)) \mbox{ is type } i,
\\ 
\paths(k) \not\in
G_{N-1} \setminus \{\paths(0)\},\; \overline{\paths(k)\paths(k+1)} 
\in \Fgasket_N,\  k=0,\cdots,L-1 \}\,. 
\arre \]
There is a natural one to one correspondence of the set $\Paths$
between different $N$s' by a scale transformation; $N$-dependence comes
only from the possible differences in the parameters $(a,b)$.
Hence we omit $N$ from the notation $\Paths(a,b,i)$.
\par
For $i \in \edges$ and $\paths \in \Paths(a,b,i)$,
let $L_i(\paths)$ be the number of steps in $\paths$ 
(ordered pairs of the form $(\paths(i),\paths(i+1))$) which are of type $i$.
Define
\[ \FF_{i}(a,b;u) \defd \sum_{\paths\in\Paths(a,b,i)} 
\prod_{i\in\edges} {u_{i}}^{L_i(\paths)}\,,\ \ \ u\in \duals\,. \]
In view of previous remarks, there is no $N$-dependence in $\FF_{i}$.
$\FF_{i}$ is a generating function of number of steps
of walks, hence is a rational function of $u$.
\par
Let $\Pi(w)={}^t(\Pi_{Ap}(w),\cdots,\Pi_{Er}(w))$ be as in \tblu{transition}.
%
\tblb[hbt] \caption{Transition probabilities} \tbla{transition} 
\begin{center} \tabb{|c||c|c|c|c|c|c|c|c|} \hline
$i$ & $Ap$ & $Ar$ & $Bp$ & $Bq$ & $Br$ & $Dq$ & $Ep$ & $Er$ \\ \hline 
$\Pi_{i}(w)$ & $\frac{w}{2+2w}$ & $\frac{1}{2+2w}$ & $ \frac{w}{1+3w}$ & 
$ \frac{w}{1+3w}$ & $ \frac{1}{1+3w}$ & $ \frac{1}{2}$ & $ \frac{w}{1+w}$ & 
$ \frac{1}{1+w}$ \\ \hline \tabe \end{center}
\tble
%
The random walk $X_{N,w_N}$ on $G_N$ defined in \secu{s1} is specified by a
positive number $w_N$, defined in \eqnu{asymp} and \eqnu{wrec}.
It is easy to see from \figu{type} that
the (one-step) jump probability of $X_{N,w_N}$ for a jump of 
type $i$ is $\Pi_{i}(w_N)$.
The definitions of $\Pi$ and $\FF$ together with \eqnu{wrec} imply 
\eqnb \eqna{PiF}
\Pi(f_{(a,b)}(w)) = \FF(a,b;\Pi(w)) \,.
\eqne
Define $\sharp\edges$-dimensional matrix $A(a,b,w)$ by
\[ A(a,b,w)_{ij} 
\defd \frac{\partial \FF_{i}}{\partial u_j}(a,b;u=\Pi(w)) \,. \]
It turns out that $A(a,b,w)_{Dq,j}$ 
for $j\ne Ap$, $Ar$, $Dq$ diverge as $w\to 0$.
We therefore define $\sharp \edges$-dimensional diagonal matrix 
$S(w) = \mbox{{\rm diag}}( S_i(w),\ i\in\edges )$ by
\eqnb \eqna{S}
S_{Dq}(w) = w \,,\ \ \  S_{i}(w) = 1 \,,\ i\ne Dq \,, 
\eqne
and define rational functions $\tFF_{i}(a,b,w;u)$, $i\in\edges$,
$u\in \duals$, by
\eqnb \eqna{tFF} 
\tFF_{i}(a,b,w;u) \defd 
S_{i}(f_{(a,b)}(w)) \, \FF_{i}(a,b;S^{-1}(w) u) \,.
\eqne
Also we define a vector $\tPi(w)$ and a matrix $\tA(a,b,w)$ by
\eqsb
\tPi(w) &\defd& S(w) \, \Pi(w)\,, \\
\tA(a,b,w)_{ij} &\defd& \AHFA\dsp
\frac{\partial \tFF_{i}}{\partial u_j}(a,b,w;u=\tPi(w)) 
= S_i(f_{(a,b)}(w)) \, A(a,b,w)_{ij} \, S_j^{-1}(w) \,.
\eqse
%
\prpb \prpa{REDUCE}
Let $a$, $a'$, $b$, and $b'$ be integers no less than $2$,
and let $I$ be the interval defined in \eqnu{asymp}.
Then the elements of matrix $\tA(a,b,w)$ are positive for $w\in I$ and 
\eqnb \eqna{supA}
\sup_{w\in I} \tA(a,b,w)_{ij} < \infty \,,\ \ \ i\in\edges,\ j\in\edges \,,
\eqne
\eqnb \eqna{infA}
\inf_{w,w'\in I} \left( \tA(a',b',w') \tA(a,b,w) \right)_{ij} > 0 \,, \ \ 
i\in\edges,\ j\in\edges \,,
\eqne
\eqnb \eqna{A22}
\tA(a,b,0)_{Ar,Ar} \ge (a+1)^2 \,.
\eqne
For each $i\in\edges$ put 
$\gg_{a,b,i}(w,h) = \tFF_{i}(a,b,w;\tPi(w)+w\, h)$.
Then $\gg_{a,b,i}$ is a rational function
in $h\in\duals$ and $w$, 
analytic at $h=0$ for $w\in I$,
and for each $j_1,\cdots,j_4 \in\edges$,
\[ \sup_{w\in I}
\left| \frac{1}{w} \frac{\partial^2 \gg_{a,b,j_1}}{\partial h_{j_2}
\partial h_{j_3} }(w,h=0) \right| < \infty\,, \ \ \mbox{and }\ 
\sup_{w\in I}
\left| \frac{1}{w} \frac{\partial^3 \gg_{a,b,j_1}}{\partial h_{j_2}
\partial h_{j_3} \partial h_{j_4}}(w,h=0) \right| < \infty\,. \]
\prpe
\prfb
$A(a,b,w)$ has non-negative elements because it is an
expectation matrix for number of steps.
 From graphical considerations
we see that every type $j$ of steps appear with positive probability
for any $i$, hence they are positive.
$\tFF$ is related to the generating function for number of steps 
of random walks (see also \prpu{branching} below), from which we see that
$\gg_{a,b,i}(w,h)$ are rational functions both in $h$ and $w$,
and analytic at $h=0$.
The parameter $w$ is the relative jump rate of the random walk.
Therefore for $w \in I$ there are no singularities.
The only possible relevant singularities of $\tFF$ are at $w=0$.
The estimates in the statement are proved by explicit calculation of
$\tFF$ with aid of computer.
We give in \appu{sb} explicit form of $\tA(a,b,w=0)$ obtained 
as the first derivatives of $\tFF$ using REDUCE.
The explicit formula implies \eqnu{supA}, \eqnu{infA}, and \eqnu{A22}.
The estimates on higher derivatives of $\tFF$ at $w=0$ is
also obtained using REDUCE.  
See \appu{sb} for more information on computer aided proof of this 
Proposition.
\prfe
 
Note that \eqnu{supA} and \eqnu{infA} imply
\eqnb \eqna{sumApos}
\inf_{w\in I} \sum_{k\in\edges} \tA(a,b,w)_{kj}>0,\ \ j\in\edges\,. 
\eqne
We go back to the gasket and look into the $N$-dependence.
Assume that $\zeta \defd \{ (a_N,b_N),\ N\in\integers \}$ 
is a bounded sequence of pairs of integers satisfying \eqnu{simple},
which determines the gasket $G$.
For $N\in\integers$, let $\vF_N=(\FF_{N,Ap},\cdots,\FF_{N,Er})$ be
a $\duals$-valued function in $\sharp \edges \, (=8)$ variables defined by
\eqnb \eqna{FN} 
\FF_{N,i}(u) \defd \FF_{i}(a_N,b_N;u)\,,\ i\in\edges. 
\eqne
Also define a diagonal matrix,
$\Pi_{N} \defd \mbox{{\rm diag}}( \Pi(w_N))$.
Using $\vF_N$ and $\Pi_N$, we can write the generating functions for
the number of steps of $X_{N,w_N}$.
Let $i \in \edges$, $j \in \edges$, and $n \le N$.
Let $Z_{nNij}$ be the random 
variable which counts the number of steps of type $j \in \edges$ 
between the times $T_{n,1}(X_{N,w_N})$ and $T_{n,0}(X_{N,w_N})$,
under the condition that 
$\left(X_{N,w_N}(T_{n,0}(X_{N,w_N})),X_{N,w_N}(T_{n,1}(X_{N,w_N}))\right)$
forms an edge of type $i$ in $\Fgasket_n$.
$T_{n,i}$ is a hitting time of $G_n$ defined in \secu{s1}.
By strong Markov property of simple random walks, 
the distribution of $Z_{nNij}$ is independent of the starting point of
$X_{N,w_N}$, and
the random variables which count the number of steps of type $j \in \edges$ 
between the times $T_{n,k+1}(X)$ and $T_{n,k}(X)$, $k=0,1,2,\cdots$, 
are independent and equal in distribution to $Z_{nNij}$,
under similar conditions.
By definition, 
$\dsp Z_{nnij}=1\ (j=i),\ \ =0\ (j\ne i)$.
%
\prpb \prpa{branching}
Fix $n\in\integers$ and $i\in\edges$.
$(Z_{nNij}\,,\ j\in\edges)$, $N=n,n+1,\cdots$, is a 
multi-type branching process whose generating functions
$\vphi_{nN}=(\phi_{nN,Ap},\cdots,\phi_{nN,Er})$ defined by
$\dsp \phi_{nNi}(z) \defd \EE{\prod_{j\in\edges} z_j^{Z_{nNij}}}$,
satisfy, for $n<N$,
\eqnb \eqna{phiF}
\vphi_{nN}(z) 
=\Pi_n^{-1} (\vF_{n+1} \circ \cdots \circ \vF_N)(\Pi_N z)\,,\ 
z\in\duals\,.
\eqne
\prpe
\prfb
The strong Markov property of simple random walks and the 
finite ramifiedness of the fractal imply that 
$\{Z_{nNij}\}$ is a branching process.
In particular, 
\eqnb \eqna{phiconvo}
\vphi_{nN}(z) = \vphi_{n,N-1}(\vphi_{N-1,N}(z))\,,\ N>n,
\eqne
holds.
The definitions of $\Pi_N$ and $\vF_N$ 
imply 
\eqnb \eqna{phiF1}
\vphi_{n,n+1}(z) 
=\Pi_n^{-1} \vF_{n+1} (\Pi_{n+1} z)\,,\ 
\eqne
which, together with \eqnu{phiconvo} implies \eqnu{phiF}.
\prfe
 
\prpu{wasymp} implies that some off-spring branching rates vanish
as $N\to\infty$, hence we are considering
branching process with singular environment.
\par
For integers $n$ and $N$ satisfying $n\le N$, define 
$\sharp\edges$-dimensional matrices 
\[ \tPi_{N} \defd \mbox{{\rm diag}}( \tPi(w_N)),\  \ 
\tA_{N} \defd \tA(a_N,b_N,w_N),\ \ 
\tB_{nN} \defd A_{n+1} A_{n+2} \cdots A_{N} \,.\]
Then \eqnu{phiF}, \eqnu{PiF}, and \eqnu{wrec} imply
\eqnb \eqna{EB}
\EE{Z_{nNij}} = 
\left( \tPi^{-1}_{n} \, \tB_{nN} \, \tPi_{N} \right)_{ij} \,,\ \ 
i\in\edges,\ j\in\edges,\ N\ge n\,. 
\eqne
Elementwise positivity of 
$A_N$ and hence of $B_{nN}$ were noted in \prpu{REDUCE};
\eqnb \eqna{ABpositive}
\tA_{Nij}>0,\ \tB_{nNij}>0,\ \ i,j\in\edges,\ N>n\,. 
\eqne
%
\prpb \prpa{estimB}
\itmb
\item
For each $n_0\in\integers$ there exist positive constants $C_1$ and $C_2$
such that if $n\in\integers$ and $N\in\integers$ satisfy $N-2\ge n \ge n_0$,
then
\eqsb
\tB_{nNij} &\ge& C_1 \prod_{k=n+1}^{N} (a_k+1)^2 \,,\\
\tB_{nNij} \,\frac{w_N^2}{w_n^2} &\ge& C_2  \prod_{k=n+1}^{N}  \left(
1+\frac{b_k}{2}\right)^2 \,, \ \ i\in\edges,\ j\in\edges.
\eqse
\item
There exist limits
\[ \gamma_{ni} \defd \lim_{N\to\infty} \frac{\tB_{nNij}}{\tB_{nN1j}} 
\,,\ \ n\in\integers,\ i\in\edges,\ j\in\edges\,, \]
independent of $j$, which satisfy, for each $n_0\in\integers$,
\[ 0 < \inf_{n\ge n_0,i\in\edges} \gamma_{ni} \le 
\sup_{n\ge n_0,i\in\edges} \gamma_{ni} < \infty \,. \]
\item
For each $n_0\in\integers$ there exists a positive constant $C_3$ 
such that if integers $n$, $m$, and $N$ satisfy
$m-1\ge n\ge n_0$ and $N\ge \max\{m,n+2\}$, then
\[ \sum_{k\in\edges} \tB_{n,m-1,i,k} \, 
\sum_{k'\in\edges} \tB_{mNk'j} \le C_3 \, \tB_{nNij}\,,
\ \ \ i\in\edges,\ j\in\edges\,. \]
\itme
\prpe
\prfb
Since $\vF$ is rational in $w$, $\tA$ is also rational.
Therefore \eqnu{A22} implies $\tA_{k22} \ge (a_k+1)^2 + C_4 \, w_k$,
$k\in\integers$, where $C_4$ is a positive constant.
\prpu{wasymp} implies that $\sum_{k\ge n} w_k < \sum_{k\ge n_0} w_k < \infty$,
$n \ge n_0$, hence we obtain the first estimate for $\tB_{nNij}$.
\prpu{wasymp} implies that for each $n_0$ there exists a constant $C_5>0$
such that 
$\dsp \frac{w_N}{w_n} \ge C_5 \, \prod_{k=n+1}^{N} \delta_k^{-1}$, 
$N\ge n\ge n_0$.
With the first estimate, we have the second estimate.
The estimates \eqnu{supA} and \eqnu{infA} imply that the elementwise
positive matrices 
$\tA_{N}$, $N\in\integers$, satisfy the assumption of \thmu{Frobenius}
in \appu{sc} with $q=2$.   
\thmu{Frobenius} then implies the second assertion.
$\zeta \defd \{ (a_N,b_N),\ N\in\integers \}$ is a bounded sequence, hence 
contains finite number of distinct pairs; as far as $\zeta$ is concerned,
taking supremum or infimum 
in $N$ is taking maximum or minimum among finite possibilities.
Assume $N \ge m+2$.  Then
\[ \arrb{l} \AHFA \dsp
\tB_{nNij} \ge \sum_{k\in\edges} \tB_{n,m-1,i,k} \sum_{k'\in\edges}
\tA_{mkk'} \, \min_{k''\in\edges} \tB_{mNk''j} 
\\ \AHFA \dsp
\ge \sum_{k} \tB_{n,m-1,i,k} 
\sum_{k'} \tB_{mNk'j} \inf_{\ell\in\edges, m' \ge n_0} 
\sum_{\ell'} \tA_{m' \ell \ell'} 
\frac{ \dsp \AHFA \inf_{k''} \{\tB_{mNk''j}/\tB_{mN1j}\} }{\dsp \AHFA
\sharp \edges \sup_{k''} \{\tB_{mNk''j}/\tB_{mN1j}\} } \,.
\arre\]
It is now easy to see that the second assertion and  \eqnu{sumApos} 
imply the third assertion.  The cases $N =m$ and $N=m+1$ can be 
proved similarly.
\prfe
 
Put $\dsp W_{nNij} \defd \frac{Z_{nNij}}{\EE{Z_{nNij}}}$.
%
\prpb \prpa{higher}
Let $i\in\edges$ and $j\in\edges$.
For each $n_0\in\integers$ there exists a positive constant $C$
such that for all $N$ and $n\ge n_0$ satisfying $N\ge n+2$,
\[ \EE{W_{nNij}^2} \le C \, \tPi_{ni} \, w_n^{-1} \,.\]
Also, for each $n\in\integers$ the third moment is bounded in $N$;
$\dsp \sup_{N\ge n} \EE{W_{nNij}^3} < \infty $.
\prpe
\prfb
By taking derivatives of $\vphi_{nN}$ in \prpu{branching} we obtain
\eqsb 
\lefteqn{
\EE{W_{nNij}^2} 
} \\
&=& \{ \sum_{k_1,k_2,k_3\in\edges} \sum_{m=n+1}^N
\left(\tPi_n^{-1} \tB_{n,m-1} \right)_{i,k_1} \,
\frac{\partial^2 \tFF_{k_1} }{\partial u_{k_2} \partial u_{k_3} }
(a_m,b_m,w_m;\tPi(w_m)) \,
\\ &&\times
\left(\tB_{m,N} \tPi_N\right)_{k_2,j} 
\left(\tB_{m,N} \tPi_N\right)_{k_3,j} 
+ \left(\tPi_n^{-1} \, \tB_{n,N} \, \tPi_N\right)_{i,j} \} \,
\EE{Z_{nNij}}^{-2} \,.
\eqse
\prpu{wasymp} implies that for each $n_0$ there exists a constant $C>0$
such that 
\[ w_N \le C \, w_n\, \prod_{k=n+1}^{N} \delta_k^{-1}\,,\ \ N\ge n\ge n_0\,. \]
With \prpu{REDUCE}, \eqnu{EB}, and \prpu{estimB} 
we have the bound for the second moment.
The bound on the third moment is proved in a similar way.
\prfe
 
Let $\edges'\subset\edges$ be the set of horizontal edges;
\eqnb \eqna{horizontal}
\edges' \defd \{ Ar,\ Br,\ Er \}\,. 
\eqne
Note that \tblu{transition} and \eqnu{S} imply that there exist positive
constants $C$ and $C'$ such that for $w\in I$ we have
\eqnb \eqna{tPi}
C \le \tPi_i(w) \le C'\,,\ \ i\in\edges'\,,\ \ \ 
C\, w \le \tPi_i(w) \le C'\, w\,,\ \ i\in\edges\setminus\edges'\,.
\eqne
Define, for $N\in\pintegers$, 
\eqnb\eqna{LN}
L_N \defd \sum_{i\in\edges}\sum_{j\in\edges} \EE{Z_{0Nij}}\,.
\eqne
The next Theorem shows that $L_N$ is the appropriate scaling factor
for the random walk $X=X_{N,W_N}$ on $H_N$. Note that 
$\dsp T_{n,1}(X)-T_{n,0}(X)=\sum_{j\in\edges} Z_{nNij}$, where 
$i$ is the type of edge formed by the endpoints of $X$ in this time interval.
%
\thmb \thma{stepconverge}
Let $m\in\integers$ and $i\in\edges$.
$(W_{mNij}$, $j\in\edges)$, $N=m,m+1,\cdots$,
converges in $L_2$ (hence in probability and in law) as $N\to\infty$. 
The limit is a random vector with equal components 
$(W_{mi},\cdots,W_{mi})$, satisfying $\EE{W_{mi}}=1$ and
$\dsp \sup_{n\ge m} w_n \EE{W_{ni}^2} < \infty$.
$\dsp L_N^{-1} \sum_{j\in\edges} Z_{mNij}$ converges in probability
as $N\to\infty$ to 
$\dsp W'_{mi} \defd \frac{\gamma_{mi}\,W_{mi}}{\sum_{j,k} \EE{Z_{0mjk}} \, 
\gamma_{mk}}\,$,
with $\gamma_{mi}$ as in \prpu{estimB}.
The distribution of $W'_{mi}$ is continuous.
\thme
\prfb
As noted in \prpu{branching},
$(Z_{mNij}$, $j\in\edges)$, $N=m,m+1,\cdots$, is a branching process.
The number of descendant at time $N$ from a single ancestor of type $k$ at 
time $n$ is equal in distribution to that of $(Z_{nNkj}$, $j\in\edges)$.
Fix $j\in\edges$.
\prpu{higher} implies, with \eqnu{tPi},
\[ \sup_{n\ge m} \sup_{N\ge n+2} w_n\,\EE{W_{nNij}^2} < \infty \,. \]
Furthermore, with $\EE{W_{nNij}}=1$ and the Schwarz inequality, this implies
the uniform integrability of $W_{nNij}$;
$\dsp \limf{p} \sup_{N\ge n+2} \EE{W_{nNij};\ W_{nNij}>p} =0 $.
This with the uniform bound for $\EE{W_{nNij}^3}$ in \prpu{higher}
and the Schwarz inequality implies
\[ \limf{p} \sup_{N\ge n+2} \EE{W_{nNij}^2;\ W_{nNij}>p} =0 \,,\ \ 
n\ge m\,. \]
\prpu{estimB} with \eqnu{EB} implies that for each $n\ge m$,
$\dsp \limf{N} \frac{\EE{Z_{nNij}}}{\EE{Z_{nN1j}}}$
exists, positive and independent of $j\in\edges$. 
Hence, if we prove that $w_N\,Z_{mNij}$ diverges in probability to infinity,
then all the assumptions of \prpu{Cohn} will be satisfied,
with $e_N$ replaced by $w_{m+N}$ and $n_0=2$. 
\prpu{Cohn} then will imply that
$(W_{mNij},\ j\in\edges)$ converges in $L_2$ as $N\to\infty$,
to a random vector with equal components.
\par
By definition, $Z_{mmij}=1$ $(j=i)$ and $=0$ $(j\ne i)$.
Fix $p>0$ and $j\in\edges$, and 
define a family of sequences $\{(x_{k,n},y_{k,n})\}$ 
$0\le n\le k$, $k\in\pintegers$, by 
\eqsb
x_{k,n} &=& \max_{i'\in\edges'} 
\prb{w_{m+n_0(k)+k}\,Z_{m+n,m+n_0(k)+k,i',j}\le p} \,, \\
y_{k,n} &=& \max_{i'\in\edges\setminus\edges'} 
\prb{w_{m+n_0(k)+k}\,Z_{m+n,m+n_0(k)+k,i',j}\le p} \,,
\eqse
where $\edges'$ is defined in \eqnu{horizontal}.
$n_0(k)$ is an arbitrary function of $k$ taking non-negative integer values
(to be specified later).
Define a sequence $\{\xa_n,\ n\in\pintegers\}$ by
$\xa_n = a_{m+n}+1$, and $\{\xw_n,\ n\in\pintegers\}$ by
\[ \xw_n \defd \max_{i'\in\edges'} 
\prb{\sum_{k\in\edges\setminus\edges'}Z_{m+n-1,m+n,i',k} \ge 1} \,. \]
$\xw_n$ is the largest probability among $i'\in\edges'$ that 
the random walk $X=X_{N,w_{N}}$ with $N=m+n$ jumps off-horizontally
at least once in the time interval $[T_{N-1,1}(X)$, $T_{N-1,0}(X)]$,
under the condition that the endpoints of $X$ for this time interval
forms an edge of type $i'$ in $\Fgasket_{N-1}$.
By definition, $3 \le \inf_n \xa_n \le \sup_n \xa_n < \infty$.  Also
$0 \le \xw_n \le 1$ and is of order $w_{m+n}$, for which \prpu{wasymp} 
implies $\xw_n \le \cv1\,\delta^{-n}$ for some constant $\cv1>0$ and 
$\dsp \delta=\min_{k} \delta_k =\min_k \frac{2(1+a_k)}{2+b_k} >1$
(recall that there are only finite number of distinct pairs $(a_k,b_k)$).
 From graphical consideration we see that
\eqsb
x_{k,n} &\le& \max_{i'\in\edges'} \left\{
\prb{\sum_{j'\in\edges\setminus\edges'}Z_{m+n,m+n+1,i',j'} =0} \,
x_{k,n+1}^{\xa_{n+1}} \right. \\ && \left. 
+ \prb{\sum_{j'\in\edges\setminus\edges'}Z_{m+n,m+n+1,i',j'} \ge 1} \, 
y_{k,n+1}\right\} \,.
\eqse
We may either use $y_{k,n} \le 1$ or use $\dsp
\prb{\sum_{j'\in\edges\setminus\edges'}Z_{m+n,m+n+1,i',j'} =0} \le 1$,
to conclude that $x_{k,n}$ satisfies the inequality in the definition of
the assumption R in \secu{s3},
with $\{w_n\}$ and $\{a_n\}$ replaced by
$\{\xw_n\}$ and $\{\xa_n\}$, respectively. Similarly, we find
\eqsb
 y_{k,n} &\le& \max_{i'\in\edges\setminus\edges'} \left\{
\prb{\sum_{j'\in\edges'}Z_{m+n,m+n+1,i',j'} \ge 1} \, x_{k,n+1} \right.
\\ && \left. +\prb{\sum_{j'\in\edges'}Z_{m+n,m+n+1,i',j'} =0} \, y_{k,n+1}
\right\} \,,
\eqse
 from which we know that
$y_{k,n}$ also satisfies the inequality of the assumption R.
\par
For $k\in\pintegers$ and $i'\in\edges'$ put
\eqsb
e_{k,i'} &\defd& \EE{w_{m+n_0(k)+k} \,Z_{m+k,m+n_0(k)+k,i',j}} \,,\\
v_{k,i'} &\defd& \VV{w_{m+n_0(k)+k} \,Z_{m+k,m+n_0(k)+k,i',j}} \,.
\eqse
\prpu{estimB}, \eqnu{EB}, \eqnu{tPi}, and $b_k\ge 2$ imply
$e_{k,i'} \ge \cv2 \, w_{m+k}^2 \, 4^{n_0(k)}$, where $\cv2>0$ is a constant
independent of $n_0\in\pintegers$ and $k\in\pintegers$.
For each $k$, define $n_0(k)$ to be sufficiently large so that
$e_{k,i'} \ge 2p$ for all $k\in\pintegers$.
\prpu{Schwarz} then implies that
$\dsp \prb{w_{m+n_0(k)+k} \,Z_{m+k,m+n_0(k)+k,i,j} \le p} < 1-1/(2d)$,
where $d\le 1+2 \,v_{k,i'} \, e_{k,i'}^{-2}$.
Applying \prpu{higher} and \eqnu{tPi} we see that
$\dsp x_{k,k} < 1-\cv3\,w_{m+k} $,
where $\cv3$ is a positive constant independent of $k$.
\prpu{wasymp} implies 
$\dsp w_{m+k} \ge \cv4\,w_{m}\,\prod_{\mu=0}^{k} \delta_{m+\mu}^{-1} $
with $\delta_k=2(1+a_k)/(2+b_k)$, and for a positive constant $\cv4$
independent of $k$.  Hence 
\[ (-\log x_{k,k}) \prod_{\mu=0}^{k} \xa_{\mu}
>\cv3\,\cv4\,w_{m} \prod_{\mu=0}^k (1+b_{m+\mu}/2) \,, \]
which implies
$\dsp \liminf_{k\to\infty,\ x_{k,k}\ne 0} \left\{ (-\log x_{k,k}) 
\prod_{\mu=0}^{k} \xa_{\mu}\right\}^{1/k} \ge 2 >1$.
We see that all the assumptions in \thmu{limit} are satisfied,
with $e_N$ replaced by $w_{m+N}$, and $Z_{N,j}$ by $Z_{m,m+N,i,j}$.
\thmu{limit} then implies
$\dsp \lim_{N\to\infty} \prb{w_{N}\, Z_{mNij} \ge p} = 1\,,\ \ p>0$,
which, as we noted in the first part of the proof,
proves the convergence of $(W_{mNij},\ j\in\edges)$
to $(W_{mi},\cdots,W_{mi})$.
Weak convergence and uniform integrability imply convergence in expectations.
Therefore, from what we have proved, we obtain the statements on 
$\EE{W_{mi}}$ and $\EE{W_{mi}^2}$.
\par
Let $n\in\pintegers$.
The $j$ independence of $\gamma_{ni}$ in \prpu{estimB} implies, with 
\eqnu{EB}, 
\[ \lim_{N\to\infty} \frac{\sum_j \EE{Z_{nNij}}}{\sum_j \EE{Z_{nN1j}}} 
=\gamma_{ni}\,. \]
Also from \eqnu{EB} one sees, for $m \le n \le N$,
\[ \EE{Z_{mNij}}= \sum_{k\in\edges} \EE{Z_{mnik}} \, \EE{Z_{nNkj}} \,. \]
Hence, with \eqnu{LN}, we see that
$\dsp 
\lim_{N\to\infty} L_N^{-1} \sum_j \EE{Z_{nNij}}=
\frac{\gamma_{ni}}{\sum_{j,k} \EE{Z_{0njk}} \,\gamma_{nk}}$.
Convergence of $(W_{nNij},\ j\in\edges)$ and \prpu{total} imply that
$\dsp \frac{\sum_{j} Z_{nNij}}{\sum_{j} \EE{Z_{nNij}}}$
converges in probability to $W_{ni}$.
Therefore we have the convergence in probability of 
$\dsp L_N^{-1} \sum_{j\in\edges} Z_{nNij}$ to $W'_{ni}$.
With $\EE{W_{ni}}=1$ and \prpu{estimB}, we have
\eqnb \eqna{ewp}
\EE{W'_{ni}}=\frac{\gamma_{ni}}{\sum_{j,k} \EE{Z_{0njk}} \, \gamma_{nk}}
\ge \cv5 \, (\sum_{j,k} \EE{Z_{0njk}})^{-1}\,,
\eqne
for some positive constant $\cv5$ independent of $n\ge 0$ and $i\in\edges$.
Similarly, there exists a positive constant $\cv6$ such that
\eqnb \eqna{ewp2}
\EE{{W'_{ni}}^2} \le \cv6 \, 
(\sum_{j,k} \EE{Z_{0njk}})^{-2} \, w_n^{-1}\,,\ \ n\ge 0,\ i\in\edges\,.
\eqne
\par
Let
\[ \Phi_{n,i}(t) \defd \EE{\exp \left( \sqrt{-1} \, t \, W'_{n,i}\right)}\,,\ 
\ t\in\reals\,,\]
denote the characteristic function.
With obvious bound
$0 \le 1- \Re \Phi_{ni}(t) \le t^2 \EE{{W'_{ni}}^2} /2$ and
$|\Im \Phi_{ni}(t)-t \EE{W'_{ni}}| \le t^2 \EE{{W'_{ni}}^2} /2$, 
$t\in\reals$,
we can proceed as in the first half of the proof of \cite[(2.45)]{HHW93}, 
using random walk representation \cite[(2.30)]{HHW93} 
for $\Phi_{ni}(t)$, to obtain
\[ |\Phi_{n,i}(t)| \le 
1 - \tC_{n} \, t^2 \,,\ \ -t'_n < t < t'_n\,,\ n\ge 0,\ i\in\edges'\,,\]
with $\dsp \tC_n = \cv7 \, w_{n+1}^{-1} \min_j \EE{W'_{n+1,j}}^2$
and $\dsp t'_n=\frac{\min_j \EE{W'_{n+1,j}}}{\max_j \EE{{W'_{n+1,j}}^2}}$,
where $\cv7$ is a positive constant independent of $n\ge0$.
(Replace $6^kt$ in \cite[(2.45)]{HHW93} by $t$ and $\dsp (3/4)^{n+k}$ by 
$w_n$.)
We may use a narrower interval $(-t_n,t_n)$ with $t_n \le t'_n$
for the estimate above, in applying \thmu{cont}.
Put $t_n \defd \cv5 \cv6^{-1} \, w_n \, \sum_{j,k} \EE{Z_{njk}}\,$.
Then \eqnu{ewp} and \eqnu{ewp2} imply $t_n \le t'_n\,$.
With  \eqnu{ewp}, \prpu{wasymp}, and \prpu{estimB}, we see that
an assumption of \thmu{cont}
\[ \liminf_{k\to\infty} 
\left( t_k^2 \tC_{k} \prod_{\mu=0}^{k-n_0} (a_{n_0+\mu+1}+1) 
\right)^{1/k} > 1 \,, \]
is satisfied with $a_\mu$ replaced by $a_{n_0+\mu+1}+1$.
\par
\prpu{estimB} and \eqnu{EB} imply 
$\dsp \limf{k} t_k = \infty$, while boundedness of $\tA_{nij}$ implied in
\eqnu{supA} with \prpu{wasymp} and \eqnu{tPi} gives 
$\sup_{k\ge 0} t_k^{1/k} < \infty$.
Let $n\ge 0$ and $m\ge 0$.
 From \eqnu{EB}, \eqnu{tPi}, \prpu{estimB}, \prpu{wasymp}, and $b_k\ge 2$,
we see 
\[ \frac{t_n}{t_{n+m}} < \frac{\cv8 \, w_n}{w_{n+m} \, \min_{\ell} 
\sum_{k\in\edges'} B_{n,n+m,\ell,k}} < \cv8 \, 4^{-m} \,,\]
where $\cv8$ is  a positive constant independent of $n$ and $m$.
Therefore there exists an $m_0$ such that 
$\dsp \frac{t_n}{t_{n+m_0}}<1$ for all $n\ge 0$.
With the boundedness of $A_{nij}$ we also see 
$\dsp \inf_{n\ge 0} \frac{t_n}{t_{n+m_0}} > 0$.
Hence all the assumptions for $\{t_k\}$ in \thmu{cont} hold.
\par
Using \cite[(2.30)]{HHW93},
we can proceed with similar arguments as we did for 
$\dsp \prb{w_{m+n_0(k)+k}\,Z_{m+n,m+n_0(k)+k,i',j}\le p}$, from which
we see that $\Phi_{n,i}$ satisfies the assumption R condition 
of \thmu{cont}.
We have now proved that $\Phi_{n,i}$ satisfies all the assumption of 
\thmu{cont}, which implies that the distribution of $W'_{n,i}$ is continuous.
\prfe
 
\cvrs1 \cvrs2 \cvrs3 \cvrs4 \cvrs5 \cvrs6 \cvrs7 \cvrs8
Let $D\defd D([0,\infty);G)$ be the set of cadlag paths 
on the \rg\ $G$.
For $n\in \integers$ and $x\in G_n$
we define a family of probability measures $\Ppt{N}{x}{\cdot}$,
$N=n,n+1,\cdots$, on $D$, by $\Ppt{N}{x}{w(0)=x}=1$ and
\[ \Ppt{N}{x}{w(t_{i})=x_{i},\; i=1,2,\cdots,r}=
\prb{X_{N,w_N,x}([L_N\, t_{i}])=x_{i},\; i=1,\cdots,r}, \]
where $X_{N,w_N,x}$ is the random walk $X_{N,w_N}$ with starting point $x$;
$X_{N,w_N,x}=x$.
We use abbreviations such as
\[ \Ppt{N}{x}{w(0)=x} \defd \Ppt{N}{x}{\{ w \in D \mid w(0)=x\} }\,, \]
and write $\Ept{N}{x}{\cdot}$ for the expectations with respect to 
$\Ppt{N}{x}{\cdot}$.
Define $T_{n,i}(w)$, $w\in D$, similarly as we did in \secu{s1} for processes,
and put $W_{n,i} \defd T_{n,i+1}-T_{n,i}$.
Let $N \ge n$, $x\in G_{N}$, $i\in\pintegers$, and let 
$x_{0},x_{1},\cdots,x_{i}$ be a sequence of points in $G_n$
such that each adjoining pair is an $n$-neighbor pair and
$x_{0}$ and $x$ are in a unit triangle of $\Fgasket_n$.
Consider the distribution of $W_{n,j}$, $j=0,1,\cdots,i-1$, 
under the conditional probability $\Ppt{N}{x}{\cdot \mid w(T_{n,j}(w))=x_j,
\; j=0,\cdots,i}$.
Since the probability is based on random walks,
this distribution is a direct product of the distributions 
of each $W_{n,j}$, and 
as we noted before \thmu{stepconverge}, 
the distribution of each $W_{n,j}$
under the conditional probability is equal to that of
$L_N^{-1} \sum_{\ell\in\edges}  Z_{nNk\ell}$, if $(x_{j},x_{j+1})$ forms 
an edge of type $k\in\edges$, and is 
independent of $i$, $j$, $x$, and $x_j$'s.
We denote this distribution of $W_{n,j}$ by  $\Qpt{N}{n,k}{\cdot}$,
and their limit distributions as $N \to \infty $ 
by $\Qlmt{n,k}{\cdot}$, $k\in\edges$.
\thmu{stepconverge} implies
\eqnb \eqna{contQ}
\limf{N} \Qpt{N}{n,k}{s\mid a<s<b}=\Qlmt{n,k}{s\mid a<s<b}\,,\ \ 
0 \le a<b \le \infty \,.
\eqne
We need a following type of uniformity to handle processes starting from
^^ irrational' points.
\prpb\prpa{tochuuten}
Let $N$, $M$, $n$ be non-negative integers satisfying $N\ge M \ge n$, 
and let $x\in G_{M}$ and $y \in G_{n}$ such that $x$ and $y$ are in 
a unit triangle of $\Fgasket_n$.
Then there exists a positive constant $\cv1$ 
independent of $x$, $y$, $n$, $M$, and $M$, such that
\[ \Ept{N}{x}{T_{n,0}(w) \mid w(T_{n,0}(w))=y }
\le \cv1 \prod_{\ell=1}^{n}  \left(1+\frac{b_{\ell}}{2}\right)^{-2}\,.\]
\prpe
\prfb
By similar arguments for the proof of \cite[(3.2)]{HHW93}, we see that
there exist positive constants $\cv2$ and $\cv3$ such that
for $X_m = X_{m,w_m,x'}$
\eqnb \eqna{time1}
\EE{T_{m-1,0}(X_m)\mid X_m(T_{m-1,0}(X_m))=y'}\le \cv1+\frac{\cv2}{w_{m}}\,,
\eqne
for all $m\in \pintegers$, $x'\in G_{m} \setminus G_{m-1}$,
and $y' \in G_{m-1}$ , with $x'$ and $y'$ in a unit triangle of 
$\Fgasket_{m-1}$.
For an $m$-neighbor pair $(u,v)$ forming a type $k$ edge,
\prpu{estimB}, \eqnu{LN}, and \eqnu{EB} imply
\eqnb \eqna{time3} \arrb{l} \dsp
L_N^{-1} \EE{W_{m,i}(X_{N,w_N})\mid X_{N,w_N}(T_{m,i})=u, 
X_{N,w_N}(T_{m,i+1})=v}
\\ \dsp
=L_N^{-1} \sum_{\ell\in\edges}  Z_{nNk\ell}\le \cv3 \, \tPi_{nk}^{-1} \, 
(\sum_{\ell,\ell'} \tB_{0.n-1,\ell,\ell'})^{-1}\,,
\arre \eqne
where $\cv3$ is a positive constant independent of 
$m$, $N$, $u$ and $v$.
\prpu{estimB}, \eqnu{time1}, and \eqnu{time3} imply
\eqnb \eqna{time4}
\Ept{N}{x'}{T_{m-1,0}(w)\mid w(T_{m-1,0}(w))=y'}\le 
\cv4\, \prod_{\ell=1}^m (1+b_\ell/2)^{-2} \,,
\eqne
for all $N\ge m\ge 0$, $x'\in G_{m} \setminus G_{m-1}$, and $y' \in G_{m-1}$, 
with $x'$ and $y'$ in a unit triangle of $\Fgasket_{m-1}$.
$\cv4$ is a constant.
The estimate \eqnu{time4}, combined with the strong Markov property of 
the random walks, implies for $N \ge M\ge n$, $x\in G_{M}$, $y\in G_{n}$,
with $x$ and $y$ in a unit triangle of $\Fgasket_{n}$,
\[ \arrb{l} \dsp
\Ept{N}{x}{T_{n,0}(w)\mid w(T_{n,0}(w))=y}
\\ \dsp
=\sum_{\{y_{i}\}} 
\sum_{i=n+1}^{M} \Ept{N}{y_i}{T_{i-1,0}(w)\mid w(T_{i-1,0}(w))=y_{i-1}}
\\ \dsp
{} \times \Ppt{N}{x}{w(T_{i,0}(w))=y_{i},\ n+1 \le i \le M-1\mid w(T_{n,0})=y}
\\ \dsp
\le \cv1 \prod_{\ell=1}^n (1+b_\ell/2)^{-2} \,,
\arre \]
where the first summation is taken over 
$\{y_{i}\}=(y_{n}, y_{n+1}, \cdots, y_{M})$
with $y_{i}\in G_{i}$, $y_{n}=y$, $y_{M}=x$, 
such that $y_{i}$ and $y_{i-1}$ are
in a unit triangle of $\Fgasket_{i-1}$, for $i=n+1, \cdots, M$.
\prfe
 
The following result is used to prove that an $N$-decimated walk of a
diffusion, obtained as the continuum limit $N\to\infty$ of a sequence of
random walks, is equal to the original random walk.
\prpb\prpa{decimated}
For $N\in\pintegers$, let $X_N$ be a simple random walk on $G_N$ with 
$N$-neighbor jumps. Assume that there exists a sequence $L_N$ diverging to
infinity as $N\to\infty$ such that, $\tX_N(\cdot) \defd X_N([L_N \, \cdot])$
converges almost surely as $N\to\infty$ to some continuous strong Markov 
process $X(\cdot)$ on $G$. Let $n\in\pintegers$. If for each $N\ge n$,
the $n$-decimated walk (defined in \secu{s1}) of $X_N$ is equal in law
to $X_n$, then the $n$-decimated walk of $X$ is also equal in law to $X_n$.
\prpe
\prfb
Fix $x\in G_n$ and $y\in G_n$. 
Denote by $\Ppt{x}{}{\cdot}$ the conditional probability with condition
$X_N(0)=x$, $N\in\pintegers$, $X(0)=x$,
and let $\Ept{x}{}{\cdot}$ be expectation with respect to $\Ppt{x}{}{\cdot}$.
For $N\ge n$ and a positive integer $q$, define
$\sigma_{N,q} \defd \inf\{t\ge 0 \mid d(\tX_N(t),G_n\setminus \{x\})\le 1/q\}$,
$\sigma_{q} \defd \inf\{t\ge 0 \mid d(X(t),G_n\setminus \{x\}) \le 1/q\}$,
$\sigma_{N,\infty} \defd \inf\{t\ge 0 \mid \tX_N(t)\in G_n\setminus \{x\} \}$,
$\sigma_{\infty} \defd \inf\{t\ge 0 \mid X(t)\in G_n\setminus \{x\} \}$,
where $d$ is the metric on $G$.
The almost sure convergence of $\tX_N$ to $X$ implies
\eqnb \eqna{qnq}
\sigma_q \le\liminf_{N\to\infty} \sigma_{N,q} \le \limsup_{N\to\infty} 
\sigma_{N,q} \le \sigma_{q+1}\,,\ a.s.,\ \ \ q>0\,.
\eqne
Define a harmonic function $h:\ G\to[0,1]$ as follows;
for $z\in G_{\infty}$, i.e., $z\in G_m$ for some $m\ge n$,
define $h(z) \defd \prb{\tX_m(\sigma_{m,\infty})=y \mid \tX_m(0)=z}$.
The assumption on the decimation property implies that 
$\prb{\tX_{m'}(\sigma_{m',\infty})=y \mid \tX_{m'}(0)=z}$ is
constant for $m'\ge m$, hence $h$ is well-defined on $G_{\infty}$.
We can see that \cite[Proposition 3.2]{HHW93} holds in our case, 
which implies that $h$ is continuous.  In particular, $h$ is uniquely 
extendable as continuous function to $G$.
By definition, $h(y)=1$ and $h(y')=0$, $y'\in G_n\setminus\{y\}$.
$X_N$ is a simple random walk, and $h$, restricted on $G_N$, is an
associated harmonic function. 
Therefore $h(\tX_N(t \wedge \sigma_{N,q}))$, $t\ge 0$, 
is a martingale ($a \wedge b \defd \min\{a,b\}$),
hence $\Ept{x}{}{h(\tX_N(t \wedge \sigma_{N,q}))}=h(x)$, $N\ge n$.
This with \eqnu{qnq}, $\limf{N} \tX_N =X$, and continuity of $h$ implies
\[ \Ept{x}{}{\min_{\sigma_q \le s \le \sigma_{q+1}} h(X(t \wedge s))} \le
h(x) \le \Ept{x}{}{\max_{\sigma_q \le s \le \sigma_{q+1}} h(X(t \wedge s))}\,.\]
Continuity of $X$ implies that $\limf{q} \sigma_{q} = \sigma_{\infty}$.
Hence we have
$h(x)= \Ept{x}{}{h(X(t \wedge \sigma_{\infty}))}$, $t\ge 0$.
Since this is independent of $t$, we have
$h(x)=\Ept{x}{}{h(X(\sigma_{\infty}))}=\prb{X(\sigma_{\infty})=y \mid X(0)=x}$,
which implies that the transition probability of $n$-decimated walk of $X$ is
equal to that of $X_n(0)$.  
\prfe
 
\prfofb{\thmu{main}}
We can apply \cite[Sect. 3]{HHW93},
with \cite[Theorems 2.5, 2.8]{HHW93} replaced by \thmu{stepconverge},
\cite[(3.1)]{HHW93} by \eqnu{contQ}, and
\cite[Proposition 3.1(1)]{HHW93} by \prpu{tochuuten}.
Then for $x_N\in G_N$, $N\in\pintegers$, satisfying $\limf{N} x_N=x$,
the sequence of measures $\Ppt{N}{x_N}{\cdot}$
(the distribution of $X_{N,w_N,x_N}([L_N t])$), $N\in\integers$,
converges weakly as $N\to\infty$ to a symmetric Feller process $X$.
Skorokhod's Theorem implies that there exists a probability space
and $G_N$ valued processes $X_N$, $N\in\integers$, such that 
$X_N$ is equal in law to $X_{N,w_N,x_N}$ and converges almost surely to 
a process equal in law to $X$.
\prpu{decimated} implies that the $n$-decimated walk of this process
is equal in law to the original random walk $X_{n,w_n}$.
That this random walk has the asymptotically one-dimensional 
(and isotropy restoration) properties, is proved in \prpu{wasymp}.
\prfofe
 
 
%----------------------------------------------------------------------
\bigskip\par\noindent {\em Acknowledgements.} 
The author would like to thank 
Prof.\ K.~Hattori and Prof.\ H.~Watanabe for collaborations
on which the present work stands, and 
Prof.\ S.~Kusuoka for the proof of \prpu{decimated}.
He would also like to thank 
Prof.\ M.~T.~Barlow and Prof.\ H.~Tasaki for valuable 
discussions concerning the present work, 
Prof.\ C.~Burdzy for the hospitality and discussions during the author's 
stay at University of Washington, Prof.\ T.~Nakagawa for bringing
references on branching processes to the author's attention,
and Prof.\ B.~M.~Hambly, Prof.\ H.~Osada, Prof.\ T.~Kumagai, 
and Prof.\ Z.~Vondra\v{c}ek, for kindly sending related 
works before publication.
 
\appendix
%----------------------------------------------------------------------
\section{Metric for scale-irregular $abb$-gaskets.} \seca{sa}
% rnda (v940528)
 
We define a metric $d$ for a \rg.
Consider a sequence of vertices of pre-gaskets, $G_0 \subset G_1 \subset
G_2 \subset \cdots$, and let $\dsp G_{\infty}=\bigcup G_n$, as in \secu{s1}.
Let $x\in G_{\infty}$ and $y\in G_{\infty}$, i.e. $x\in G_N$, $y\in G_{N}$,
for some $N$.
We explain the case  where $x$ and $y$ are in a unit triangle of $G_0$.
The case where $x$ and $y$ are ^^ further apart' can be handled similarly.
\par
Let $path(x,y)$ be a collection of finite sequences
$z=\{z_0=x,z_1,\cdots,z_K=y\}$ for some $K=K(z)\ge 0$,
which has a property that
for each $i=0,1,\cdots,K(z)-1$, there exists $s_z(i)\in\pintegers$
such that $z_i$ and $z_{i+1}$ are ${s_z(i)}$-neighbor pairs.
Let $\{a_N,b_N\}$ be a bounded sequence of pair of positive integers,
that determines $G_{\infty}$.
Put $\dsp \lambda_n=\prod_{k=1}^{n} \min\{a_k+1,b_k+1\}^{-1}$, 
$n\in\pintegers$.
For $z\in path(x,y)$ put $\dsp L(z)=\sum_{i=0}^{K(z)-1} \lambda_{s_z(i)}$.
Then the metric $d$ is defined to be
$\dsp d(x,y) \defd \inf_{z\in path(x,y)} L(z)$.
It is straightforward to see that 
if $x$ and $y$ are an $N$-neighbor pair, then 
\eqnb \eqna{dxy} d(x,y)=\lambda_N\,. \eqne
That $d$ is a metric is also easy to see.
\par
The definition of $\lambda_n$ can be replaced by
$\dsp \lambda_n=\prod_{k=1}^{n} \ell_k^{-1}$, 
for any set of $\ell_k$ satisfying
$\ell_k \le \min\{a_k+1,b_k+1\}$
and 
$\inf_k \ell_k >1$.
The first condition implies \eqnu{dxy}.  The second condition 
with the boundedness of $\{a_N,b_N\}$ implies that there exists a
positive constant $C$ such that if $x$ and $y$ is in a unit triangle
of $G_N$ then $d(x,y)\le C\,\lambda_N\,$.
This property means that our definition of $d$ is compatible with our analysis
in this paper based on the hierarchical (recursive) structure of the fractal.
 
%----------------------------------------------------------------------
\section{Decay estimates from non-linear recursion relations.} \seca{sd}
% rndd (v940528)
 
The Lemma below gives a mild decay estimate
 from a non-linear recursion relation.
We apply the Lemma to prove a Theorem which states a sharp decay estimate
 from another recursion relation with more involved assumptions.
\lemb \lema{recineq}
Let $\{w_n,\ n\in\pintegers\}$ be a sequence in $[0,1]$
satisfying $\sum_n w_n < \infty$, and 
$\{a_n,\ n\in\pintegers\}$ a sequence satisfying
$ D\defd \inf_n a_n>1$ and $\sup_n a_n < \infty$.
For each $k\in\pintegers$ define a sequence $\{x_{k,n},\ n=k,k-1,\cdots,0\}$
by a recursion relation
\[ x_{k,n}=(1-w_{n+1}) \, x_{k,n+1}^{a_{n+1}}+w_{n+1}\,,\ \ 
n=k-1,k-2,\cdots,0\,,\]
with initial condition $x_{k,k}$ satisfying $0 \le x_{k,k} \le 1$.
If 
\[ \lim_{k\to\infty,\ x_{k,k}\ne 0} 
(-\log x_{k,k}) \prod_{\mu=0}^{k} a_{\mu} = \infty \]
holds, then there exist positive constants $C_1$ and $k_1$ 
(independent of $n$ and $k$) such that
\[ x_{k,n} \le C_1 \sup_{\mu \ge n} w_{\mu} 
+ \exp\left(-D^{f_k-n-1}\right) \,,\ \ 0\le n \le f_k,\ k\ge k_1\,,\]
where 
\[ f_k=\sup \{ n \le k-1 \mid \sqrt{D} \, (-\log x_{k,k}) 
\prod_{\mu=n+2}^{k} a_{\mu} > 1 \} +1 \,, \]
with a convention $\prod_{\mu=k+1}^{k} a_{\mu} = 1$,
and $f_k=k$ if $x_{k,k}=0$.
\leme
\prfb
Put $C_2 = \sup_n \exp(a_n)$. $\sum_n w_n < \infty$ and $D>1$ imply that 
there exists a constant $n_1$ such that
$\dsp \prod_{\mu \ge n} (1-C_2 \, w_{\mu}) \ge \sqrt{D}^{-1} $, $n\ge n_1\,$.
By assumption, $\dsp \lim_{k\to\infty} f_k =\infty$, hence there exists 
a constant $k_1$ such that $\dsp f_k \ge n_1 -1$, $k \ge k_1$.
Let $k \ge k_1$ in the following. We first prove that 
\eqnb \eqna{step1}
x_{k,f_k} \le \exp\left( - D^{-1} \right) \,,\ \ \ k \ge k_1\,.
\eqne
If $x_{k,k}=0$ then \eqnu{step1} directly follows, so we assume $x_{k,k}>0$.
Put $u_{k,n}= - \log x_{k,n}$. The assumptions and the recursion relation 
imply $ 0< x_{k,n} \le 1 $ for all $n$, hence $u_{k,n}$ exist and 
are non-negative. Furthermore,
\[ u_{k,n}=a_{n+1} u_{k,n+1} - \log \left(1+ ( x_{k,n+1}^{-a_{n+1}}-1) \, 
w_{n+1}\right) \le a_{n+1} u_{k,n+1} \,, \]
which implies
\eqnb \eqna{d0}
u_{k,n} \le u_{k,k} \left( \prod_{\mu=n+1}^{k} a_{\mu} \right) \,,\ \ 
0\le n\le k\,.
\eqne
The definition of $f_k$ implies $f_k \le k$ and the following 
three inequalities;
\eqnb \eqna{d1}
u_{k,k} \ge \sqrt{D}^{-1} \,,\
\ \ \mbox{if } f_k = k \,,
\eqne
\eqnb \eqna{d2}
\sqrt{D} \, u_{k,k} \left( \prod_{\mu=f_k+1}^{k} a_{\mu}\right)  > 1\,,
\eqne
\eqnb \eqna{d3}
\sqrt{D} \, u_{k,k} \left( \prod_{\mu=n+2}^{k} a_{\mu}\right)  \le 1 \,, 
\ \ f_k \le n \le k-1\,.
\eqne
 From \eqnu{d0}, \eqnu{d3}, and $D>1$ follows
$ u_{k,n+1} < 1 $, $ f_k \le n \le k-1$,
which, together with the recursion relation implies
\[ \arrb{l} 
u_{k,n}=a_{n+1} u_{k,n+1} - \log (1+\exp(u_{k,n+1} a_{n+1})\, 
w_{n+1} (1-\exp(-u_{k,n+1} a_{n+1}))) 
\\
\ge a_{n+1} u_{k,n+1}-\log\left(1+C_2\, w_{n+1}\, u_{k,n+1}\, a_{n+1}\right)
\\
\ge a_{n+1} (1 - C_2 w_{n+1}) \, u_{k,n+1}\,,
\ \ \ \ f_k \le n \le k-1\,.
\arre \]
This with $k \ge k_1$ and \eqnu{d2} implies
$u_{k,f_k} \ge
u_{k,k} ( \prod_{\mu=f_k+1}^{k} a_{\mu} ) 
( \prod_{\mu=f_k+1}^{k} (1 - C_2 w_{\mu}) ) \ge D^{-1} $,
if $f_k \le k-1$.
If $f_k \ge k$ then $f_k =k $, hence \eqnu{d1} implies 
$\dsp u_{k,f_k} > D^{-1}$.
Therefore we have \eqnu{step1}.
\par
Put $\dsp v_n = \sup_{\mu \ge n} w_{\mu}$.
$\{v_n\}$ is decreasing, bounded above by $1$, and
$\dsp \lim_{n\to\infty} v_n = 0$.
Define a sequence $\{z_n,\ n=f_k,f_k-1,\cdots,0\}$ by
$z_{k,n} = z_{k,n+1}^D + v_{n+1} \,$, $0\le n\le f_k-1$,
and $z_{k,f_k}=\exp(-D^{-1})$.
Then 
\eqnb \eqna{d6}
x_{k,n} \le z_{k,n}\,,\ \ \ \ 0\le n \le f_k\,.
\eqne
Put 
\eqnb \eqna{d7}
z_{k,n}= \exp\left( -D^{f_k-n-1}\right) + v_n \, r_{k,n} \,. 
\eqne
Taylor's Theorem implies, with $D>1$,
\[ r_{k,n} \le \frac{v_{n+1}}{v_n} \, r_{k,n+1} \, D 
\left( \exp\left( -D^{f_k-n-2} \right) + v_{n+1} \, r_{k,n+1} \right)^{D-1} 
+ v_{n+1} \,, \]
which further is reduced with the properties of $v_n$ to
\eqnb \eqna{d5}
r_{k,n} \le r_{k,n+1} \, D \, (e^{-1} + v_{n+1}  \, r_{k,n+1} )^{D-1} +1 \,,
\ \ \ 0 \le n \le f_k-2\,.
\eqne
Put $\dsp \rho=\frac{1}{2} \, (1+D \, e^{-D+1})$.
$D>1$ implies $0< D \, e^{-D+1} < \rho <1$.
Therefore there exists a constant $k_2$ defined by
\[ k_2 = \inf \{ n\ge 0 \mid D \, (e^{-1} + v_n \, (1-\rho)^{-1} )^{D-1}
< \rho \} \,. \]
Monotonicity of $\{v_n\}$ implies 
$\dsp D\, (e^{-1} + v_n \, (1-\rho)^{-1} )^{D-1} < \rho$, $n \ge k_2\,$.
If $f_k \ge k_2 +1$ then we can prove by induction that
$\dsp r_{k,n} \le (1-\rho)^{-1} $, $k_2 \le n \le f_k\,$.
In fact, we explicitly have $\dsp r_{k,f_k}=0$  and 
$\dsp r_{k,f_k-1}= v_{f_k-2}/ v_{f_k-1} \le 1$.
If $\dsp r_{k,n+1} \le (1-\rho)^{-1} $
holds for $n$ some with $k_2 \le n \le f_k-2$,
then \eqnu{d5} and the definition of $k_2$ implies 
$\dsp r_{k,n} \le (1-\rho)^{-1}\rho +1 = (1-\rho)^{-1}$.
Thus if $f_k \ge k_2 +1$, 
$r_{k,n}$ for $k_2 \le n \le f_k$ are bounded 
by a constant independent of $n$ and $k$.
$k_2$ is independent of $n$ and $k$.
Therefore $r_{k,n}$ for $0 \le n \le k_2$ are bounded 
by a constant independent of $n$ and $k$.
If $f_k < k_2 +1$, similar argument shows, with $r_{k,f_k}=0$, that 
$r_{k,n}$ for $0 \le n \le f_k$ are bounded 
by a finite number independent of $n$ and $k$.
This with \eqnu{d6} and \eqnu{d7} implies the statement.
\prfe
 
\thmb \thma{recineq}
Let $\{w_n\}$, $\{w'_n\}$, $n\in\pintegers$, be sequences in $[0,1]$
satisfying 
\[ \max\{w_n,\ w'_n\} \le C_w \delta^{-n} \,, \ \ \ n\in\pintegers \,, \]
for positive constants (independent of $n$) 
$C_w$ and $\delta > 1$.
Also let $\{a_n,\ n\in\pintegers\}$ be a sequence satisfying
$ \inf_n a_n \ge 2$ and $\sup_n a_n < \infty$.
For each $k\in\pintegers$ consider a sequence in $[0,1]^2$
\[ \{(x_{k,n},y_{k,n})\,,\ \ n=k,k-1,\cdots,0\} \ \subset [0,1]^2 \,, \]
and assume that it satisfies a recursive inequality
\eqsb
x_{k,n} &\le& x_{k,n+1}^{a_{n+1}}+w_{n+1} \, 
\min\{1-x_{k,n+1}^{a_{n+1}}\,,\ y_{k,n+1}\} \,,\ \ \\
y_{k,n} &\le& x_{k,n+1}+ w'_{n+1} \, y_{k,n+1} \,,\ \ 
n=k-1,k-2,\cdots,0\,.
\eqse
If 
\eqnb \eqna{deltap}
\liminf_{k\to\infty,\ x_{k,k}\ne 0} 
\left\{ (-\log x_{k,k}) \prod_{\mu=0}^{k} a_{\mu}\right\}^{1/k} >1 
\eqne
holds, then there exist positive constants $C_1$ and $C_2$
(independent of $k$) such that
\[ \max\{x_{k,0},\ y_{k,0}\} \le 
C_1 \, \exp(-C_2 \, k^2) \,,\ \ \ k\in\pintegers \,. \]
\thme
\prfb
Define $\{\tilde{x}_{k,n},\ n=k,k-1,\cdots,0 \} $ by
$\tilde{x}_{k,k}=x_{k,k}$ and
\[ \tilde{x}_{k,n}=(1-w_{n+1}) \, \tilde{x}_{k,n+1}^{a_{n+1}}+w_{n+1}\,,\ \ 
n=k-1,k-2,\cdots,0\,,\]
Then the recursion relation for $x_{k,n}$ and the assumption $y_{k,n} \le 1$
imply 
\[ x_{k,n} \le \tilde{x}_{k,n} \,,\ \ \ 0 \le n \le k,\ k\ge 0\,. \]
$\{\tilde{x}_{k,n} \}$ satisfies all the assumptions
of \lemu{recineq} with $D=2$, hence
there exist positive constants $C_3$ and $k_1$ 
(independent of $n$ and $k$) such that 
\eqnb \eqna{x}
x_{k,n} \le C_3 \sup_{\mu \ge n} w_{\mu} 
+ \exp\left(-2^{f_k-n-1}\right) \,,\ \ 0\le n \le f_k,\ k\ge k_1\,,
\eqne
where 
\eqnb \eqna{fk}
f_k=\sup \{ n \le k-1 \mid \sqrt{2} \, (-\log x_{k,k}) 
\prod_{\mu=n+2}^{k} a_{\mu} > 1 \} +1 \,. 
\eqne
Since \eqnu{deltap} implies
$\dsp \lim_{k\to\infty} f_k = \infty$,
there exists a constant $k_2 \ge k_1$ such that 
for $0 \le n \le f_k /2$ and $k \ge k_2$,
\[ 
\exp\left(-2^{f_k-n-1}\right) \le 
\exp\left(-2^{f_k/2-1}\right) \le 
C_w \delta^{-f_k/2} \le
C_w \delta^{-n} \,.
\]
This with \eqnu{x} implies
$\dsp x_{k,n} \le (C_3+1) \, C_w \, \delta^{-n} $,
$0 \le n \le f_k /2 $, $k \ge k_2 $.
Applying this estimate to the original recursion relations and using
$w_n \le C_w \delta^{-n}$, $w'_n \le C_w \delta^{-n}$, and $a_n \ge 2$, we have
\eqsb
x_{k,n} &\le& C_w \,\delta^{-n-1} \, ( (C_3+1)\, x_{k,n+1} + y_{k,n+1}) \,,\ \ 
\\
y_{k,n} &\le& x_{k,n+1}+ C_w \, \delta^{-n-1} \, y_{k,n+1} \,,\ \ 
0 \le n \le f_k /2 \,,\ k \ge k_2 \,.
\eqse
Iterating once, we find
\[ \max\{ x_{k,n},\ y_{k,n} \} 
\le C_5 \, \delta^{-n-1} \, \max \{ x_{k,n+2},\ y_{k,n+2} \} 
\,,\ \ \ 0 \le n \le f_k /2 -1\,,\ k \ge k_2 \,,
\]
where $C_5$ is a positive constant independent of $n$ and $k$.
Iterating this $[f_k /4]$ times, where $[x]$ is the largest integer
not exceeding $x$, and using $x_{k,n} \le 1$, $y_{k,n} \le 1$,
we find
\eqnb \eqna{xy}
{}\ \ \ \ \max\{ x_{k,0},\ y_{k,0} \} 
\le \exp \left\{ \left[\frac{f_k}{4}\right] \, (\log C_5)
- \left[\frac{f_k}{4}\right]^2 \, (\log \delta) \right\} \,,\ \ k\ge k_2\,.
\eqne
The assumption \eqnu{deltap}
implies that there exist positive constants $k_3 \ge k_2$ and
$\delta' > 1$ (independent of $k$) such that 
$\dsp (-\log x_{k,k}) \prod_{\mu=0}^{k} a_{\mu} 
> \delta'^k$, $k\ge k_3$.
The definition \eqnu{fk} then implies
$\dsp f_k \ge \min \left\{ \frac{\log \delta'}{\log \sup_{\mu} a_{\mu}} \; k 
-1 ,\ k \right\}$, $ k\ge k_3$.
Applying this to \eqnu{xy}, increasing constants for terms with $k<k_3$
if necessary, we have the statement.
\prfe
 
%----------------------------------------------------------------------
\section{Products of matrices with positive elements.} \seca{sc}
% rndc (v940523)
 
We present an elementary theorem on the existence of a limit of normalized 
products of matrices with positive elements.
We assume no relation among matrices in the product, such as
commutativity or stationarity.
We also allow the infimums of some components to be zero.
\thmb \thma{Frobenius}
Let $d$ and $q$ be positive integers, $\edges \defd \{1,2,\cdots,d \}$, and
$\{A_N$, $N=1,2,3,\cdots\}$ be a sequence of $d$-dimensional matrices
whose elements are positive and bounded, satisfying
$\dsp \inf_{N,i,j} \left( A_N A_{N+1} \cdots A_{N+q-1} \right)_{ij} > 0 $.
Then for $i\in\edges$ and $j\in\edges$,
$\dsp \gamma_{i} \defd \lim_{N\to\infty} 
\frac{\dsp \left( A_1 \cdots A_{N} \right)_{ij}}{\dsp
\left( A_1 \cdots A_{N} \right)_{1j}} $
exists, positive, and is independent of $j$.
\thme
\prfb
For $N>n\ge0$ define
\[ B_{nN} \defd A_{n+1} A_{n+2} \cdots A_{N} \]
and put
\[ \gamma_{Nij} \defd \frac{B_{0Nij}}{B_{0N1j}} \,,\ \ 
i\in\edges,\ j\in\edges\,. \]
The elementwise positivity of $A_{N+1}$ and $B_{nN}$ imply 
for each $i$ that 
$\dsp \{ \min_k \gamma_{Nik} \}$ is increasing and 
$\dsp \{ \max_k \gamma_{Nik} \}$ is decreasing in $N$,
in particular, the sequence $\{\gamma_{Nij},\ N =1,2,\cdots\}$ is compact.
Therefore, for each $i$ and $j$, and
for any subsequence of positive integers there exists
a further subsequence $\{a_N\}$ such that the limit
\eqnb \eqna{agammapre}
\gamma_{ij}^{(a)} \defd \lim_{N\to\infty} \gamma_{a_N,ij} > 0\,.
\eqne
exists and is positive.
\par
For $0 < n < N$ and $i\in\edges$, $j\in\edges$, put 
\[ p_{nNij}\defd \frac{B_{0n1i} \, B_{nNij}}{B_{0N1j}} \,. \]
The definition and the elementwise positivity of $B_{nNij}$ imply,
for $0 < n < N$, $i,j\in\edges$,
\eqnb \eqna{pprob}
0 < p_{nNij} < 1\,,\ \ \sum_{k\in\edges} p_{nNkj}=1 \,,\ \ 
\gamma_{Nij} = \sum_{k\in\edges} \gamma_{nik} \, p_{nNkj}\,.
\eqne
We prove a couple of Lemma for $p_{nNkj}$.
\lemb \lema{l1}
Fix $\{a_N\}$, and let $\gamma_{ij}^{(a)}$ be as above.
If for every $i,j\in\edges$ either
$\dsp \inf_{n>0} \inf_{N> n+q} p_{nNij} > 0$ or
$\dsp \inf_{n>0} \inf_{N> n+q} p_{nNji} > 0$ hold,
then for every $i\in\edges$, $\gamma_{ij}^{(a)}$ is independent of $j$.
\leme
\prfb
Put $n=a_M$ and $N=a_{M'}$ in \eqnu{pprob}.
We see from \eqnu{agammapre} and \eqnu{pprob} that for each 
$\epsilon>0$ there is an integer $M_0$ such that for any integers $M$, $M'$
satisfying $M'>M>M_0$ we have
\eqnb \eqna{jgamma}
\left| \sum_{k\in\edges} (\gamma_{ij}^{(a)}-\gamma_{ik}^{(a)}) 
p_{a_M,a_{M'},kj} \right| < \epsilon \,. 
\eqne
Now suppose that the Lemma is wrong;
$\dsp \gamma_{ik_1}^{(a)} < \gamma_{ik_2}^{(a)}$
and
$\dsp \gamma_{ik_1}^{(a)} \le \gamma_{ij}^{(a)} \le \gamma_{ik_2}^{(a)}$,
$ j\in\edges$.
If we put $j=k_2$ in \eqnu{jgamma} and keep $k=k_1$ term in the summation
we have
$\dsp \epsilon > (\gamma_{ik_2}^{(a)}-\gamma_{ik_1}^{(a)}) 
p_{a_M,a_{M'},k_2,k_1} $,
while if we put $j=k_1$ and keep $k=k_2$ term we have
$\dsp \epsilon > (\gamma_{ik_2}^{(a)}-\gamma_{ik_1}^{(a)}) 
p_{a_M,a_{M'},k_1,k_2} $.
Since $\epsilon>0$ is arbitrary, these inequalities contradicts the
assumption of the Lemma.
\prfe
 
\lemb \lema{l2}
\[ \inf_{n>0} \inf_{N>n+q} p_{nNij} > 0\,,\ \ \ i\in\edges,\ j\in\edges\,. \]
\leme
\prfb
Note that each $p_{nNij}$ is positive by \eqnu{pprob}.  
Therefore it is sufficient to consider the cases
where $N$ and $n$ are sufficiently large.
For sufficiently large $N$,
\[ \arrb{l} \dsp
p_{1Nij} 
\ge \frac{A_{11i}}{\dsp \sum_{k_1} A_{11k_1}} \,
\frac{\dsp \sum_{k_2} A_{2,i,k_2} \, B_{2,N,k_2,j}}{\dsp \max_{k_1} 
\sum_{k_2} A_{2,k_1,k_2} \, B_{2,N,k_2,j}}
\ge \frac{A_{11i}}{\dsp \sum_{k_1} A_{11k_1}} \, 
\min_{k_2} \frac{\dsp A_{2,i,k_2}}{\dsp \max_{k_1} A_{2,k_1,k_2}}\,,
\arre\]
where we used an inequality among non-negative numbers
$a_i$, $b_i$, $c_i$, $i\in\edges$;
$\dsp \frac{\sum a_i c_i}{\sum b_i c_i} \ge \min_i \frac{a_i}{b_i}\,$.
Hence 
$\dsp \inf_{N>1} p_{1Nij} > 0 $.
If we prove
$\dsp  \inf_{n>0} \inf_{N> n+q}  \frac{p_{nNij}}{p_{1Nij}} > 0 $,
then the Lemma is proved.
For sufficiently large $N$ and $n$ with $N-q > n$,
\[ \arrb{l} \dsp
\frac{p_{nNij}}{p_{1Nij}}
\ge \frac{\dsp \min_k A_{11k} B_{1nki} B_{nNij}}{\dsp 
\max_{k} A_{11i} B_{1nik} B_{nNkj}}
\ge \min_{\{k_i\}}
\frac{\dsp \min_k A_{11k} A_{2kk_1} B_{n-q,n,k_2i} B_{n,n+q,ik_3}}{\dsp
A_{11i} A_{2ik_1} \max_k B_{n-q,n,k_2k} B_{n,n+q,kk_3}}\,.
\arre\]
Taking the infimum of both sides with respect to $N$ and $n$, we see, 
with the assumptions of \thmu{Frobenius},
$\dsp \inf_{n>0} \inf_{N> n+q} \frac{p_{nNij}}{p_{1Nij}} > 0 $.
\prfe
 
Let us continue the proof of the Theorem.
\lemu{l1} and \lemu{l2} imply that $\gamma_{ij}^{(a)}$ of \eqnu{agammapre} 
is independent of $j$.
Fix $i\in\edges$, and 
consider two subsequences of positive integers.
There are subsubsequences, $\{a_N\}$ and $\{b_N\}$, for each of the
subsequences respectively, such that the limits
\eqnb \eqna{agamma}
\gamma_{i}^{(a)} \defd \lim_{N\to\infty} \gamma_{a_N,ij} > 0\,, \ \ 
\mbox{and }\ 
\gamma_{i}^{(b)} \defd \lim_{N\to\infty} \gamma_{b_N,ij} > 0\,,
\eqne
exist, positive, and are independent of $j$.
Put $n=b_M$ and $N=a_{M'}$ in \eqnu{pprob};
\eqnb \eqna{abSD}
\gamma_{a_{M'},ij} = \sum_{k\in\edges} \gamma_{b_M,ik} \, p_{b_M,a_{M'},kj}\,,
\ \ \ a_{M'} > b_M \,,\ i\in\edges,\ j\in\edges \,.
\eqne
The equations \eqnu{agamma}, \eqnu{abSD}, and \eqnu{pprob}
imply that for any positive $\epsilon$ there exists an integer $N_0$ 
such that if $a_{M'}> b_M >N_0$ hold, then
\[ \left| \gamma_{i}^{(a)}-\gamma_{i}^{(b)} \right|
= \left| \gamma_{i}^{(a)}-\left(\gamma_{i}^{(b)}
\sum_{k\in\edges} p_{b_M,a_{M'},kj}\right)  \right| < \epsilon\,,\ j\in\edges\,.\]
Hence $\gamma_{i}^{(a)}=\gamma_{i}^{(b)}$, which implies that
the limit is independent of subsequences.
Positivity of the limit also follows from \eqnu{agamma}.
\prfe
 
%----------------------------------------------------------------------
\section{Basic estimates on generating functions.} \seca{sb}
% rndb (v940527)
% hh1 = \tilde{A}(a,b,w=0),  h1 = hh1 (b+2)^2/(a+1),  M(a,b)_{ij} = h1
% h1 from delta2 19931214(tue)
 
We give an explicit formula for the generating function
\[\gg_{a,b,i}(w,h) = \tFF_{i}(a,b,w;\tPi(w)+w\, h)\,,\] 
introduced in \secu{s2}.
For each $i\in\edges$, $\gg_{a,b,i}(w,h)=Num_i / Den_i$, where
\[ Den_i = \det W + \sum_{\alpha=1}^{3} \det \left(\arrb{c|c} 
0 & O_{\alpha} \\ \hline I_{\alpha,i} & W
\arre \right) \,, \]
\[ Num_i = - \det \left(\arrb{c|c} 
0 & O'_i \\ \hline I'_i & W
\arre \right) \,. \]
Put $\dsp Z(i) \defd \Pi(w)_{i} + w\, S^{-1}(w)_{ii} \,h_i$, $i\in\edges$.
Then $O_1 = (Z(Ar),0,0,0,0,Z(Bq))$, $O_2 = (0,Z(Ar),Z(Bq),0,0,0)$, 
and $O_3 = (0,0,0,Z(Bp),Z(Bp),0)$.
For $X\in\{A,B,D,E\}$ and $t\in\{p,q,r\}$,
we write $Xt$ to specify an element in $\edges$, with an obvious rule.
With this convention, $O'_{Xp} = O_3$, $X\in\{A,B,E\}$, $O'_{Dq} = O_1$, and 
$O'_i = O_2$, otherwise.
$I_{1,Dq} = 0$, otherwise $I_{1,Xt} = {}^t (Z(Xr),0,0,0,0,Z(Xp))$.
$I_{2,At} = {}^t (0,Z(Ar),Z(Ap),0,0,0)$, otherwise $I_{2,Xt} = 0$.
$I_{3,Xt} = 0$, if $X\in\{A,E\}$, otherwise 
$I_{3,Xt} = {}^t (0,0,0,Z(Xq),Z(Xq),0)$.
$I'_{Xq} = {}^t I_{3,Xq}$, while for $t \ne q$, 
$I'_{Xt} = I_{1,Xt}$.
$W$ is a $6$ dimensional matrix given by
\[ \arrb{l} W= {\rm I} - \\ \left( \arrb{cccccc}
 W_{\alpha}(1) & W_{\beta}(1) & 0&0&0& Z(Bq) \\
 W_{\beta}(1) & W_{\alpha}(1) & Z(Bq) &0&0&0 \\
0& Z(Ap) & W_{\alpha}(2) & W_{\beta}'(2) &0&0 \\
0&0& W_{\beta}(2) & W_{{\alpha}'}(2) & Z(Br) &0 \\
0&0&0& Z(Br) & W_{{\alpha}'}(2) & W_{\beta}(2) \\
Z(Ap) &0&0&0& W_{{\beta}'}(2) & W_{\alpha}(2) 
\arre \right) \,.\arre \]
For $j=1,2$,
$\dsp W_{\alpha}(j)+{\alpha}'(j)=W_{{\alpha}'}(j)+{\alpha}(j)=
1-\bar{{\beta}}(j) \, 
\frac{\Delta(j)_{n(j)}}{\Delta(j)_{n(j)-1}}\,$,
$\dsp W_{\beta}(j)=\frac{\bar{{\beta}}(j)}{\Delta(j)_{n(j)-1}} \, 
\left(\frac{{\beta}(j)}{{\beta}'(j)}\right)^{n(j)/2}$,
$\dsp W_{{\beta}'}(j)=\frac{\bar{{\beta}}(j)}{\Delta(j)_{n(j)-1}} \, 
\left(\frac{{\beta}'(j)}{{\beta}(j)}\right)^{n(j)/2}$,
where
$\dsp \bar{{\beta}}(j)=\sqrt{{\beta}(j)\,{\beta}'(j)}\,$, 
$2\bar{{\alpha}}(j)={\alpha}(j)+{\alpha}'(j)$, and
$\dsp \Delta(j)_{n(j)} 
= \frac{x_+(j)^{n(j)+1} -x_-(j)^{n(j)+1}}{x_+(j) -x_-(j)}$,
$\dsp x_{\pm}(j) = 1+\delta(j) \pm \sqrt{\delta(j) \, (2+\delta(j))}$,
$\dsp \delta(j) = \frac{1-2\bar{{\alpha}}(j)
-2\bar{{\beta}}(j)}{2\bar{{\beta}}(j)}\,$. 
Finally, $n(j)$, ${\alpha}(j)$, ${\alpha}'(j)$, ${\beta}(j)$, 
and ${\beta}'(j)$ are given by 
$n(1)=a-1$, $n(2)=b-1$, 
${\alpha}(1)={\alpha}'(1)=Z(Ap)\, Z(Dq)$, ${\alpha}(2)=Z(Br)\, Z(Er)$, 
${\alpha}'(2)=Z(Bq)\, Z(Ep)$, 
${\beta}(1)={\beta}'(1)=Z(Ar)+Z(Ap)\, Z(Dq) $, 
${\beta}(2)=Z(Bp)+Z(Br)\, Z(Ep) $, 
${\beta}'(2)=Z(Bq)+Z(Bq)\, Z(Er) $.
\par
With these explicit formula, we obtain the following order estimate.
Define $C_i$, $i\in\edges$, by $C_{Ar}=1/2$, $C_{Br}=C_{Er}=1$, and $C_i=0$,
otherwise.
\prpb \prpa{3}
For all $i\in\edges$, $\dsp w^{-3} \, Den_i$ and
$\dsp w^{-4} \, (Num_i -C_i\, Den_i)$ are rational 
in $w$ and $h$, analytic at $w=h=0$.
\prpe
 
We also find by REDUCE calculation that $O(w^3)$ terms in $Den_i$ do not 
vanish;
\eqnb \eqna{novanish}
\lim_{w\to 0} w^{-3} \, Den_i \ne 0 \,,\ \ \ i\in\edges.
\eqne
 
The matrix $\tA(a,b,w)$ defined in \secu{s2} 
is rational in $w$, and has no poles in $w\ge 0$.
The explicit form of $\tA(a,b,w=0)$ given below is obtained by explicit
calculation of the first derivatives of $\tFF$ given above, using REDUCE.
\par
Define, for notational simplicity, a matrix $M(a,b)$ by
$M(a,b)_{ij} = (b+2)^2 (a+1)^{-1} \tA(a,b,w=0)_{ij}$,
$i,j \in \edges $,
and put $B_2 = b+2$.  Then
\[ M(a,b)=\left[ \arrb{ccc}
 2 B_2 &
 0 &
 (b^3 + 9 b^2 + 14 b + 12)/12 
\\
 (a B_2 + b) B_2 &
 (a + 1) B_2^2 &
 (b^2 + 4 b + 6) (b - 1)/6 
\\
 2 B_2 &
 0 &
 (b^3 + 9 b^2 + 20 b + 24)/6 
\\
 0 &
 0 &
 b (b + 4) (b - 1)/6 
\\
 2 (a B_2 - 1) B_2 &
 2 a B_2^2 &
 b (b^2 + 4 b + 7) 
\\
 0 &
 0 &
 b (b + 1) B_2/4 
\\
 2 B_2 &
 0 &
 (b^3 + 9 b^2 + 14 b + 12)/6 
\\
 2 (a B_2 - 1) B_2 &
 2 a B_2^2 &
 (b^2 + 4 b + 6) (b - 1)/3 
\arre \right. \]

\[ \left. \arrb{ccc}
 b (b + 4) (b - 1)/6 &
 b (b + 5) (b + 1)/12 &
 0 
\\
 b (b^2 + 6 b + 11)/3 &
 b (2 b^2 + 9 b + 13)/12 &
 (a - 1) B_2^2 
\\
 b (b + 4) (b - 1)/3 &
 b (b + 5) (b + 1)/6 &
 0 
\\
 (b^3 + 9 b^2 + 14 b + 12)/3 &
 b (b + 5) (b + 1)/6 &
 0 
\\
 2 b (b^2 + 3 b + 5) &
 (2 b^2 + 5 b + 8) (b + 1)/2 &
 2 (a - 1) B_2^2 
\\
 b (b + 1) B_2/2 &
 b (b + 1) B_2/4 &
 B_2^2 
\\
 b (b + 4) (b - 1)/3 &
 b (b + 5) (b + 1)/6 &
 0 
\\
 2 b (b^2 + 6 b + 11)/3 &
 b (2 b^2 + 9 b + 13)/6 &
 2 (a - 1) B_2^2 
\arre \right. \]

\[ \left. \arrb{cc}
 (b^2 + 10 b + 12) (b - 1)/12 &
 b (b + 7) (b - 1)/12 
\\
 (b^2 + 4 b + 6) (b - 1)/6 &
 (2 b^2 + 11 b + 24) (b - 1)/12 
\\
 (b^2 + 10 b + 12) (b - 1)/6 &
 b (b + 7) (b - 1)/6 
\\
 b (b + 4) (b - 1)/6 &
 b (b + 7) (b - 1)/6 
\\
 (b^2 + 2 b + 2) (b - 1) &
 (2 b^2 + 3 b + 8) (b - 1)/2 
\\
 b (b - 1) B_2/4 &
 b (b - 1) B_2/4 
\\
 b (b + 8) (b + 1)/6 &
 b (b + 7) (b - 1)/6 
\\
 b (b^2 + 6 b + 11)/3 &
 b (2 b^2 + 15 b + 37)/6 
\arre \right] \ . \]
\par
All the formula in this appendix except \eqnu{novanish} and the explicit 
form of $\tA$ are derived by hand.  
At present the only proof of 
\eqnu{novanish} is by REDUCE, but it is a ^^ soft' result.
It is straightforward to see that \prpu{3} and \eqnu{novanish} imply the
estimates in \prpu{REDUCE} for second and third derivatives of $\gg$.
The explicit form of $\tA$ is also derived by REDUCE.
The required estimates in \secu{s2} are \eqnu{supA}, \eqnu{infA}, and 
\eqnu{A22}, among which \eqnu{supA} and \eqnu{infA} reflects the
network structure of the (pre-) fractal, and \eqnu{supA} is an expectation
related to one-dimensional simple random walk.
It therefore suffices with relatively soft estimates of $\tA$.
With these considerations, presumably, we may be able to avoid computer
aided proof after all.
For our purpose, rigorous derivation by REDUCE is sufficient.

Technical methods of derivation for the formula and estimates 
derived by hand in this appendix may be interesting in it's own respect,
but the derivations are lengthy, 
and taking into account of the main interest of this paper,
we omit further explanations.


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