On the rate of convergence in the central limit theorem for hierarchical Laplacian

TL;DR

This paper investigates the rate at which the distribution of normalized eigenvalue means of a perturbed hierarchical Laplacian on ultrametric spaces converges to a normal distribution, providing estimates in total variation distance.

Contribution

It extends previous CLT results for hierarchical Laplacians by quantifying the convergence rate in total variation distance under mild assumptions.

Findings

Normalized eigenvalue means converge in law to a normal distribution.

Provides explicit estimates for the rate of convergence in total variation distance.

Enhances understanding of spectral properties of perturbed hierarchical Laplacians.

Abstract

Let $(X,d)$ be a proper ultrametric space. Given a measure $m$ on $X$ and a function $C(B)$ defined on the set of all non-singleton balls $B$ we consider the hierarchical Laplacian $L=L_{C}$. Choosing a sequence $\{\varepsilon (B)\}$ of i.i.d. random variables we define the perturbed function $C(B,\omega )$ and the perturbed hierarchical Laplacian $L^{\omega }=L_{C(\omega )}.$ We study the arithmetic means $\overline{\lambda }(\omega )$ of the $L^{\omega }$-eigenvalues. Under some mild assumptions the normalized arithmetic means $\big( \overline{\lambda }-\mathbb{E}\overline{\lambda }\big) /\sigma \big( \overline{\lambda }\big) $ converge in law to the standard normal distribution. In this note we study convergence in the total variation distance and estimate the rate of convergence.

Equations

$$d(x,y)\le \max \{d(x,z),d(z,y)\}.$$

$$\lambda (B)$$

$$\lambda (\{a\})$$

$$L_{C}f(x):=\sum _{B\in \mathcal{B}:B\ni x}C(B)\left(f(x)-\frac{1}{m(B)}{\int }_{B}fdm\right).$$

$$f_B=\frac{1_B}{m(B)}-\frac{1_{B^{\prime }}}{m(B^{\prime })}.$$

$$m(B)=\{\begin{array}{ll}{n}_1{n}_2\cdots n_k& \text{for k>0, and}\\ 1/(n_{k+1}{n}_{k+2}\cdots n_0)& \text{for k<0 in Case 1.}\end{array}$$

$$\lambda (B)={\lambda }_k\text{ for all }B\in {\mathcal{B}}_k,\text{where}{\lambda }_k=\sum _{\ell \ge k}{C}_{\ell }.$$

$$G=\bigcup _k{G}_k,\text{and}{n}_k=[G_k:G_{k-1}]$$

$$\beginpicture\par\setcoordinatesystem units<.8mm,1.3mm>\par\setplotarea xfrom-10to104,yfrom-14to46\par\arrow<6pt>[.2,.67]from22to4040\par\plot 3232622/\par\plot 1616302/\par\plot 4816342/\par\plot 88142/\par\plot 248182/\par\plot 408462/\par\plot 568502/\par\plot 4462/\par\plot 124102/\par\plot 204222/\par\plot 284262/\par\plot 364382/\par\plot 444422/\par\plot 524542/\par\plot 604582/\par\par\par\arrow<6pt>[.2,.67]from9929to8840\par\plot 6629632/\par\plot 702684/\par\plot 742764/\par\plot 782728/\par\plot 822888/\par\plot 862844/\par\plot 902924/\par\plot 9428016/\par\par\setdots<3pt>\putrule from-4.84to1024\putrule from-4.58to1028\putrule from-216to10216\putrule from-1.732to10232\par\setdashes<2pt>.5pt\putrule from-2-5to102-5\par\put(0.0,0.0){$}andtakingvaluesinsomesmallinterval$[-\epsilon,\epsilon]\subset\left( -1,1\right) .$Wedefinetheperturbedfunction$C(B,\omega)$andtheperturbedhierarchicalLaplacianasfollows:\begin{equation*}C(B,\omega)=C(B)(1+\varepsilon(B,\omega))\end{equation*}and\begin{equation*}L^{\omega}f(x)=L_{C(\omega)}f(x)=\sum\limits_{B\in\mathcal{B}:\text{ }x\in B}C(B,\omega)\left(f(x)-\frac{1}{m(B)}\int_{B}f\,dm\right).\end{equation*}Evidently$L^{\omega}$maywellbenon-homogeneousforsome$\omega\in\Omega$.Stillithasapurepointspectrumforall$\omega$butthestructureoftheclosedset$\mathsf{Spec}(L^{\omega})$canbequitecomplicated,see\cite[cite]{[\@@bibref{}{BendikovKrupski}{}{}]}forvariousexamples.\par Letusfixahorocyle$H$andcomputetheeigenvalue$\lambda(B,\omega)$for$B$in$H$.Withoutlossofgeneralitywemayassumethat$H=H_{0}$.Let$\varpi$betheAlexandrovpointandlet$\{B_{k}\}_{k\geq 0}$betheuniqueinfinitegeodesicpathin$\mathcal{T}(X)$from$B$to$\varpi$.Wehave\begin{aligned} \lambda(B,\omega)&=\sum\limits_{k\geq 0}C(B_{k},\omega)=\sum\limits_{k\geq 0}C_{k}\left(1+\varepsilon(B_{k},\omega)\right)\\ &=\lambda_{0}\left(1+\sum\limits_{k\geq 0}a_{k}\varepsilon(B_{k},\omega)\right)=\lambda_{0}\left(1+U(B,\omega)\right),\end{aligned}where$a_{k}=C_{k}/\lambda_{0},$and\begin{equation}U(B,\omega)=\sum\limits_{k\geq 0}a_{k}\varepsilon(B_{k},\omega).\end{equation}Noticethat$\sum_{k\geq 0}a_{k}=1$andthat$\{U(B)\}_{B\in H}$are(dependent)identicallydistributedsymmetricrandomvariablestakingvaluesinsomesymmetricinterval$I\subsetneqq\left( -1,1\right) .$Inparticular,$\mathbb{E}\lambda(B,\cdot)=\lambda_{0}$.\par\par\@@unnumbered@section{paragraph}{toc}{Normal approximation.} \par Let us choose a reference point $o\in X,$ say the neutral element in our group-identification, and let $\mathcal{O}$ denote the family of all balls $O$ centered at $o.$ Let $\mathcal{B}_{0}(O),O\in\mathcal{O},$ be the set of all balls $B$ in $O$ each of which belongs to the horocycle $H_{0}$. We set \begin{equation*}\overline{\lambda}_{O}(\omega)=\frac{1}{\left|\mathcal{B}_{0}(O)\right|}\sum\limits_{B\in\mathcal{B}_{0}(O)}\lambda(B,\omega)\end{equation*}and \begin{equation*}\Lambda_{O}\left(\omega\right)=\frac{\overline{\lambda}_{O}(\omega)-\lambda_{0}}{\sigma\left(\overline{\lambda}_{O}\right)},\end{equation*}where $\left|\mathcal{B}_{0}(O)\right|$ stands for cardinality of the finite set $\mathcal{B}_{0}(O)$. \par According to \cite[cite]{[\@@bibref{}{BGMS}{}{}, Theorem 3.2]}, as $O\rightarrow\varpi,$ $\Lambda_{O}$ converges in law to the standard normal random variable $Z$ whenever the following condition holds \begin{equation}1/\kappa\leq C(B)\left(\mathsf{diam}(B)\right)^{\delta/2}\leq\kappa,\end{equation}or equivalently, \begin{equation}1/2\kappa\leq\lambda(B)\left(\mathsf{diam}(B)\right)^{\delta/2}\leq 2\kappa,\end{equation}for some $\kappa>0$ and $\delta\geq 1$, \par The main aim of this paper is to strengthen this result, namely we want to prove that under certain conditions the convergence $\Lambda_{O}\to Z$ holds in the total variation distance, see Theorem \ref{Thm_approx}. \par\par\@@unnumbered@section{paragraph}{toc}{Metric matters.} \par Given two probability distributions $P$ and $Q$ on the real line the total variation distance from $P$ to $Q$ is defined as \@@amsalign\|P-Q\|_{TV}=2\!\!\!\sup_{A\in\mathrm{B}(\mathbb{R})}\Big{|}P(A)-Q(A)\Big{|}. Notice that if $P$ and $Q$ are absolutely continuous with respect to some measure $\theta$ and $f$ and $g$ are their densities then \@@amsalign\|P-Q\|_{TV}=2\|f-g\|_{L^{1}(\text{d}\theta)}. The relative entropy (also called the Kullback-Leibler distance) of $P$ with respect to $Q$ is defined as \@@amsalign D(P\!\parallel\!Q)=\int_{\mathbb{R}}\log_{2}\left(\frac{\text{d}P}{\text{d}Q}\right)\text{d}P, when $P$ is absolutely continuous with respect to $Q$ and $D(P\!\!\parallel\!\!Q)=+\infty$ otherwise. \par Let $X$ be a random variable with the density $p$ with respect to the Lebesgue measure. Its differential entropy is defined as \@@amsalign h(X)=-\int_{\mathbb{R}}p(x)\log_{2}p(x)\,\text{d}x. We also consider the following quantity \@@amsalign D(X)=h(W)-h(X)=\int_{\mathbb{R}}p(x)\log_{2}\frac{p(x)}{q(x)}\,\text{d}x, where $W$ is a normal random variable with the density $q$ (with resp. to the Lebesgue measure) such that $\mathbb{E}(W)=\mathbb{E}(X)$ and $\mathrm{Var}(W)=\mathrm{Var}(X)$. Observe that $D(X)=D(P_{X}\!\parallel\!P_{W})$, where $P_{X}$ denotes the distribution of the random variable $X$. We have $D(a+bX)=D(X)$, $0\neq b,a\in\mathbb{R}$, and thus $D$ is mean and variance invariant. Moreover, if $P_{X}$ is equal to $N(0,\sigma^{2})$ then \@@amsalign h(X)=\frac{\log_{2}(2\pi e\sigma^{2})}{2}\quad\mathrm{and}\quad D(X)=0. \par We recall that by the Pinsker inequality \cite[cite]{[\@@bibref{}{Pinsker}{}{}]}, the entropic distance dominates the total variation, that is \@@amsalign\|P_{X}-P_{W}\|_{TV}^{2}\leq 2D(X). \par We shall need the following bound on the relative entropy expressed in terms of densities \cite[cite]{[\@@bibref{}{Bobkov}{}{}, Lemma 2.2]}. Let $X$ be a random variable whose distribution function $F_{X}(x)$ is absolutely continuous with $F_{X}^{\prime}(x)=$ $p(x),$ assume further that the first absolute moment of $X$ is finite. Let $Z$ be the standard normal random variable and $F_{Z}^{\prime}(x)=\phi(x)$ be its density. For any $T\geq 0$ we have \begin{equation}\begin{split}D(P_{X}\!\parallel\!P_{Z})&\leq e^{-T^{2}/2}+\sqrt{2\pi}\int_{-T}^{T}\left(p(x)-\phi(x)\right)^{2}e^{x^{2}/2}\text{d}x\\ &\quad+\frac{1}{2}\int_{|x|\geq T}x^{2}p(x)\text{d}x+\int_{|x|\geq T}p(x)\log_{2}p(x)\text{d}x.\end{split}\end{equation} \par\par\@@unnumbered@section{paragraph}{toc}{Berry-Essen bounds.} \par Let $X_{1},X_{2},\ldots,X_{n}$ be independent random variables with $\mathbb{E}(X_{k})=0$ and with finite variances $\sigma_{k}^{2}=\mathbb{E}(X_{k}^{2})$. Assuming that for some $s>2$ and all $1\leq k\leq n,$ $\mathbb{E}|X_{k}|^{s}<\infty$ we define the following quantities \begin{equation*}B_{n}=\sum_{k=1}^{n}\sigma_{k}^{2}\text{ ,\ \ \ \ }S_{n}=\frac{X_{1}+\ldots+X_{n}}{B_{n}^{1/2}},\end{equation*}and \begin{equation*}\mathbb{L}_{s}=\frac{1}{B_{n}^{s/2}}\sum_{k=1}^{n}\mathbb{E}|X_{k}|^{s},\end{equation*}the Lyapunov ratios. Let $P_{n}$ be the distribution of the random variable $S_{n}$ and $\mathcal{N}$ be the standard normal distribution. One of the main ingredients in our analysis is the following result, see {\cite[cite]{[\@@bibref{}{Bobkov}{}{}, Theorems 1.1 and 1.2]}}. \par\begin{theorem} Assume that $D(X_{k})\leq A$ for some positive constant $A$, and all $1\leq k\leq n.$ The following statements hold true \par\begin{enumerate} \par\enumerate@item@[1.] $\mathbb{L}_{3}<+\infty$ implies $\|P_{n}-\mathcal{N}\|_{TV}\leq C\mathbb{L}_{3},$ \par\par\enumerate@item@[2.] $\mathbb{L}_{4}<+\infty$ implies $D(S_{n})\leq C^{\prime}\mathbb{L}_{4},$ \par\end{enumerate} where the constants $C$ and $C^{\prime}$ depend only on $A$. \end{theorem} \par We shall need the following technical result which we will prove here for completeness. \par\begin{lemma} Let $X,Y$ be two independent random variables such that$\mathrm{Var}(X)+\mathrm{Var}(Y)=1$. Then \begin{equation*}D(X+Y)\leq\mathrm{Var}(X)D(X)+\mathrm{Var}(Y)D(Y).\end{equation*}In particular, for any collection of independent random variables $\{X_{k}\}$, \begin{equation*}D\left(\sum_{k=1}^{n}X_{k}\right)\leq\max_{1\leq k\leq n}D(X_{k}).\end{equation*} \end{lemma} \par\noindent{Proof.} We apply the following inequality, see \cite[cite]{[\@@bibref{}{Johnson}{}{}, Theorem D.1]}, \begin{equation}2^{2h(X)}+2^{2h(Y)}\leq 2^{2h(X+Y)}.\end{equation}According to \eqref{entropy_normal}, we have \begin{equation*}D(X+Y)=h(Z)-h(X+Y)\quad\mathrm{and}\quad h(Z)=\log_{2}\sqrt{2\pi e}.\end{equation*}Similarly, we write \@@amsalign D(X)=h(Z_{1})+h(X)\quad\mathrm{and}\quad D(Y)=h(Z_{2})+h(Y), where $Z_{1}$ and $Z_{2}$ are normal random variables such that $\mathrm{Var}(Z_{1})=\mathrm{Var}(X)$ and $\mathrm{Var}(Z_{2})=\mathrm{Var}(Y)$. Using this in \eqref{power_entropy} we get \begin{equation*}2^{h(Z_{1})}\cdot 2^{-2D(X)}+2^{h(Z_{2})}\cdot 2^{-2D(Y)}\leq 2^{2h(Z_{1}+Z_{2})}\cdot 2^{-2D(X+Y)},\end{equation*}whence \begin{equation*}\mathrm{Var}(X)\cdot 2^{-2D(X)}+\mathrm{Var}(Y)\cdot 2^{-2D(Y)}\leq 2^{-2D(X+Y)}.\end{equation*}Since $x\mapsto 2^{-2x}$ is convex, we get \begin{equation*}2^{-2\left[\mathrm{Var}(X)D(X)+\mathrm{Var}(Y)D(Y)\right]}\leq\mathrm{Var}(X)\cdot 2^{-2D(X)}+\mathrm{Var}(Y)\cdot 2^{-2D(Y)}.\end{equation*}The result follows. \rule{5.0pt}{5.0pt} \par\par\@@numbered@section{section}{toc}{Central limit theorem} \par The main result of the article is the following theorem. \par\begin{theorem} In notation of Section \ref{Prem_sec}, suppose that condition \eqref{C-B condition11} holds with $\delta\geq 1$. Assume that the random variable $\epsilon(B)$ has a bounded density with respect to the Lebesgue measure. Denote $v_{N}=n_{1}n_{2}\cdot\cdot\cdot n_{N},$ clearly $v_{N}\geq 2^{N}$. For $O\in H_{N}$ the following inequality holds \begin{equation*}\|P_{\Lambda_{O}}-\mathcal{N}\|_{TV}\leq\begin{cases}C\,N^{-\frac{1}{2}}&\text{for $\delta=1,$}\\ C\,v_{N}^{-\frac{3}{2}\min\{\delta-1,\frac{1}{3}\}}&\text{for $\delta>1$},\end{cases}\end{equation*}for some $C>0$ and all $N\geq 1$. In particular, independently on the group structure \begin{equation*}\|P_{\Lambda_{O}}-\mathcal{N}\|_{TV}\leq\begin{cases}C\,N^{-\frac{1}{2}}&\text{for $\delta=1,$}\\ C\,2^{-\frac{3}{2}N\min\{\delta-1,\frac{1}{3}\}}&\text{for $\delta>1$}.\end{cases}\end{equation*} \end{theorem} \par\noindent{Proof.} For $O\in H_{N}$ we write $\overline{\lambda}_{N}(\omega)$ for $\overline{\lambda}_{O}(\omega)$ and $\Lambda_{N}(\omega)$ for $\Lambda_{O}(\omega)$, then \@@amsalign\overline{\lambda}_{N}(\omega)&=\lambda_{0}\left(1+\overline{U}_{N}(\omega)\right),\\ \overline{U}_{N}(\omega)&=\frac{1}{v_{N}}\sum\limits_{B\in\mathcal{B}_{0}(O)}U(B,\omega),\\ \Lambda_{N}(\omega)&=\frac{\overline{U}_{N}(\omega)}{\sigma\left(\overline{U}_{N}\right)}. To estimate the quantity $\|P_{\Lambda_{O}}-\mathcal{N}\|_{TV}$ we apply Theorem \ref{THM_Berry-Essen}. We distinguish two cases {$(\delta>1)$ }and {$(\delta=1).$}\newline {The case $(\delta>1)$}: Let $\{O_{k}\}_{k\geq N}$ be the infinite geodesic path from $O$ to $\varpi$. Similarly, for each $B\in H_{0}$ we pick the infinite geodesic path $\{B_{k}\}_{k\geq 0}$ from $B$ to $\varpi$. Applying $(\ref{U-epsilon})$ to equation $(\ref{U-bar})$ we obtain\@@eqnarray\overline{U}_{N}&=&\frac{1}{n_{1}...n_{N}}\sum\limits_{B\in H_{0}:B\subset O}\sum\limits_{k\geq 0}a_{k}\varepsilon(B_{k})=\frac{1}{n_{1}...n_{N}}\sum\limits_{k\geq 0}a_{k}\sum\limits_{B\in H_{0}:B\subset O}\varepsilon(B_{k})\\ &=&\frac{1}{n_{1}...n_{N}}\left(a_{0}\sum\limits_{B_{0}\in H_{0}:B_{0}\subseteq O}\varepsilon(B_{0})+a_{1}n_{1}\sum\limits_{B_{1}\in H_{1}:B_{1}\subseteq O}\varepsilon(B_{1})\right.\\ &&\left.+a_{2}n_{1}n_{2}\sum\limits_{B_{2}\in H_{2}:B_{2}\subseteq O}\varepsilon(B_{2})+...+a_{N}n_{1}n_{2}...n_{N}\varepsilon(O_{N})\right)\\ &&\quad+a_{N+1}\varepsilon(O_{N+1})+a_{N+2}\varepsilon(O_{N+2})+...\text{ }.Let us introduce two random variables \@@amsalign\mathcal{U}_{N}=\frac{a_{0}}{n_{1}...n_{N}}\sum\limits_{B_{0}\in H_{0}:B_{0}\subseteq O}\varepsilon(B_{0})+\frac{a_{1}}{n_{2}...n_{N}}\sum\limits_{B_{1}\in H_{1}:B_{1}\subseteq O}\varepsilon(B_{1})+\ldots+a_{N}\varepsilon(O_{N}) and\begin{equation}X_{N}=a_{N+1}\varepsilon(O_{N+1})+a_{N+2}\varepsilon(O_{N+2})+...\text{ }.\end{equation}Random variables $\mathcal{U}_{N}$ and $X_{N}$ are independent, have zero mean and\begin{equation*}\overline{U}_{N}=\mathcal{U}_{N}+X_{N}.\end{equation*}Let us denote by $\mathcal{B}_{k}(O)$ the set of all balls $B\subset O$ which belong to the horocycle $H_{k}$, and let $\mathcal{B}^{\ast}(O)=\bigcup_{k=0}^{N}\mathcal{B}_{k}(O)$. Then \begin{equation*}\Lambda_{N}=\frac{\sum_{B\in\mathcal{B}^{\ast}(O)}X_{B}+X_{N}}{B_{N}^{1/2}},\end{equation*}where \begin{equation*}X_{B}=\frac{a_{k}}{n_{k+1}...n_{N}}\varepsilon(B),\text{ for }B\in\mathcal{B}_{k}(O),\end{equation*}and \begin{equation*}B_{N}=\sum_{B\in\mathcal{B}^{\ast}(O)}\sigma^{2}\left(X_{B}\right)+\sigma^{2}\left(X_{N}\right).\end{equation*} \par As $\epsilon(B)$ are bounded i.i.d. having bounded density, $D(X_{B})=D(\epsilon(B))<\infty$ for all balls $B\in\mathcal{B}^{\ast}(O)$. We claim that, for some $A>0$, \begin{equation*}D(X_{N})\leq A,\quad\mathrm{for\ all}\ N.\end{equation*}Indeed, let $\widetilde{X_{N}}=X_{N}/\sqrt{\sigma(X_{N})}$. Then $D\left(X_{N}\right)=D\big{(}\widetilde{X_{N}}\big{)}$. Random variable $\widetilde{X_{N}}$ is bounded because, by \eqref{C-B condition11}, \begin{equation*}\left|\widetilde{X_{N}}\right|\leq\frac{\sum_{k>N}a_{k}}{\sqrt{\sum_{k>N}a_{k}^{2}}}\leq C\frac{a_{N+1}}{\sqrt{a_{N+1}^{2}}}=C<\infty,\end{equation*}for some $C>0$. It follows that the density $\widetilde{p_{N}}$ of $\widetilde{X_{N}}$ is continuous and compactly supported function, whence applying \eqref{Entropy_density} we obtain \begin{equation*}D\big{(}\widetilde{X_{N}}\big{)}=D\big{(}\widetilde{X_{N}}\!\parallel\!Z\big{)}\leq e^{-\frac{C^{2}}{2}}+\sqrt{2\pi}e^{\frac{C^{2}}{2}}\|\widetilde{p_{N}}-\phi\|_{2}^{2},\end{equation*}where $\phi$ is the standard normal density. Let $\Phi_{\widetilde{X_{N}}}$ and $\Phi$ be the characteristic functions of random variables $\widetilde{X_{N}}$ and $Z$ respectively. The Plancherel formula yields \begin{equation*}\|\widetilde{p_{N}}-\phi\|_{2}^{2}=\frac{1}{2\pi}\|\Phi_{\widetilde{X_{N}}}-\Phi\|_{2}^{2}\leq\frac{1}{\pi}\left(\|\Phi_{\widetilde{X_{N}}}\|_{2}^{2}+\|\Phi\|_{2}^{2}\right).\end{equation*}Let $\Phi_{\varepsilon}$ be the characteristic function of $\varepsilon(B)$. We have \begin{equation*}|\Phi_{\widetilde{X_{N}}}(\xi)|=\prod_{k=N+1}^{\infty}\Big{|}\Phi_{\varepsilon}\left(\frac{a_{k}}{\sqrt{\sigma(X_{N})}}\xi\right)\!\Big{|}\leq\Big{|}\Phi_{\varepsilon}\left(\frac{a_{N+1}}{\sqrt{\sigma(X_{N})}}\xi\right)\!\Big{|},\end{equation*}whence \@@amsalign\|\Phi_{\widetilde{X_{N}}}\|_{2}^{2}&\leq\int\Big{|}\Phi_{\varepsilon}\left(\frac{a_{N+1}}{\sqrt{\sigma(X_{N})}}\xi\right)\Big{|}^{2}\text{d}\xi=\frac{\sqrt{\sigma(X_{N})}}{a_{N+1}}\|\Phi_{\varepsilon}\|_{2}^{2}=\frac{\sqrt{\sum_{k>N}a_{k}^{2}}}{a_{N+1}}\|\Phi_{\varepsilon}\|_{2}^{2}\\ &\leq C^{\prime}\|\Phi_{\varepsilon}\|_{2}^{2},where $C^{\prime}$ does not depend on $N$. \par Next we want to estimate the Lyapunov ratio \begin{equation*}\mathbb{L}_{s}=\frac{\sum_{B\in\mathcal{B}(O)}\mathbb{E}|X_{B}|^{s}+\mathbb{E}|X_{N}|^{s}}{B_{N}^{s/2}}.\end{equation*}Since $\mathcal{U}_{N}$ and $X_{N}$ are independent, we have \begin{equation*}B_{N}=\sigma\left(\overline{U}_{N}\right)^{2}=\sigma\left(\mathcal{U}_{N}\right)^{2}+\sigma\left(X_{N}\right)^{2}.\end{equation*} By \cite[cite]{[\@@bibref{}{BGMS}{}{}, Claim 1]}, \begin{equation*}\sigma\left(X_{N}\right)^{2}\asymp\left(n_{0}\cdots n_{N}n_{N+1}\right)^{-\delta}\end{equation*}and \begin{equation}\sigma\left(\mathcal{U}_{N}\right)^{2}\asymp\left\{\begin{array}[]{ccc}\left(n_{0}\cdots n_{N}\right)^{-1}&\text{if}&\delta>1,\\ N\cdot\left(n_{0}\cdots n_{N}\right)^{-1}&\text{if}&\delta=1,\\ \left(n_{0}\cdots n_{N}\right)^{-\delta}&\text{if}&\delta<1,\end{array}\right.\end{equation}for $N$ large enough. Hence, for $\delta\geq 1$ and $N$ big enough, \begin{equation}B_{N}\asymp\sigma\left(\mathcal{U}_{N}\right)^{2}.\end{equation}Further we have \@@amsalign\sum_{B\in\mathcal{B}_{0}(O)}\mathbb{E}|X_{B}|^{s}+\mathbb{E}|X_{N}|^{s}&=\left(\frac{a_{0}}{n_{1}\cdots n_{N}}\right)^{s}\sum\limits_{B_{0}\in H_{0}:B_{0}\subset O}\!\!\!\!\mathbb{E}|\varepsilon(B_{0})|^{s}+\left(\frac{a_{1}}{n_{2}\cdots n_{N}}\right)^{s}\!\!\!\!\sum\limits_{B_{1}\in H_{1}:B_{1}\subset O}\!\!\!\!\mathbb{E}|\varepsilon(B_{1})|^{s}\\ &\qquad+\ldots+a_{N}^{s}\mathbb{E}|\varepsilon(O_{N})|^{s}+\mathbb{E}\Big{|}\sum_{k=N+1}^{\infty}a_{k}\varepsilon(O_{k})\Big{|}^{s}\\ &\asymp\frac{1}{v_{N}^{s-1}}\left(1+v_{1}^{s-1-s\delta/2}+\ldots+v_{N}^{s-1-s\delta/2}\right)whence \begin{equation*}\sum_{B\in\mathcal{B}_{0}(O)}\mathbb{E}|X_{B}|^{s}+\mathbb{E}|X_{N}|^{s}\asymp\begin{cases}\frac{1}{v_{N}^{s-1}}&\text{if $s-1-s\delta/2<0$,}\\ \frac{1}{v_{N}^{s/2-1}}&\text{if $s-1-s\delta/2\geq 0$}.\end{cases}\end{equation*}Combining this with \eqref{B_N} we obtain \begin{equation*}\mathbb{L}_{s}\asymp\begin{cases}\frac{1}{v_{N}^{s/2-1}}&\text{if $s-1-s\delta/2<0$,}\\ \frac{1}{v_{N}^{s/2(\delta-1)}}&\text{if $s-1-s\delta/2\geq 0$.}\end{cases}\end{equation*}Thus, for $s=3$ and $\delta>1$, we get the desired result.\newline {The case} $(\delta=1)$: Let us introduce auxiliary notation \@@amsalign Y_{0}&=\frac{a_{0}}{n_{1}...n_{N}}\sum\limits_{B\in H_{0}:B\subseteq O}\varepsilon(B),\\ Y_{1}&=\frac{a_{1}}{n_{2}...n_{N}}\sum\limits_{B\in H_{1}:B\subseteq O}\varepsilon(B),\\ &\ \ldots\ \\ Y_{N}&=a_{N}\varepsilon(O_{N})\quad\mathrm{and}\quad Y_{N+1}=X_{N}.Clearly, each of $Y_{k}$ has mean zero and \begin{equation*}\Lambda_{N}=\frac{Y_{0}+Y_{1}+\ldots+Y_{N+1}}{B_{N+1}^{1/2}},\quad B_{N+1}=\sum_{k=0}^{N+1}\sigma\left(Y_{k}\right)^{2}.\end{equation*}We again apply Theorem \ref{THM_Berry-Essen} because, by Lemma \ref{lemma_entropy}, \begin{equation*}D(Y_{k})=D\left(\sum\limits_{B\in H_{k}:B\subseteq O}\varepsilon(B)\right)\leq D(\epsilon(B))<\infty,\quad 0\leq k\leq N.\end{equation*}\par To estimate the Lyapunov ratios we use the Marcinkiewicz-Zygmund inequality \cite[cite]{[\@@bibref{}{Shiryaev}{}{}, Chapter VII, \S 3]}: for independent random variables $\xi_{1},\xi_{2},\ldots,\xi_{n}$ with mean zero and for all $p>1$ there is a constant $C(p)>0$ such that \begin{equation*}\mathbb{E}\Big{|}\sum_{k=1}^{n}\xi_{k}\Big{|}^{p}\leq C(p)\,\mathbb{E}\left(\sum_{k=1}^{n}\xi_{k}^{2}\right)^{p/2}.\end{equation*}Setting \begin{equation*}\xi_{k}=\sum_{B\in H_{k}:B\subset O}\varepsilon(B),\end{equation*}the Marcinkiewicz-Zygmund inequality yields \begin{equation*}\mathbb{E}|Y_{k}|^{s}=\left(\frac{a_{k}}{n_{k+1}\cdots n_{N}}\right)^{s}\mathbb{E}|\xi_{k}|^{s}\leq C(s)\mathrm{Var}(\epsilon(B))\frac{a_{k}^{s}}{(n_{k+1}\cdots n_{N})^{s/2}}.\end{equation*}It follows, see \eqref{C-B condition11}, that \@@amsalign\sum_{k=0}^{N}\mathbb{E}|Y_{k}|^{s}&\leq C_{1}(s)\left(\frac{a_{0}^{s}}{(n_{1}\cdots n_{N})^{s/2}}+\frac{a_{1}^{s}}{(n_{2}\cdots n_{N})^{s/2}}+\ldots+a_{N}^{s}\right)\\ &=\frac{C_{1}(s)}{v_{N}^{s/2}}\left(a_{0}^{s}+a_{1}^{s}n_{1}^{s/2}+\ldots+a_{N}^{s}v_{N}^{s/2}\right)\leq C_{2}(s)\frac{N}{v_{N}^{s/2}}.Hence, using \eqref{B_N} and \eqref{U-L comparisong} with $\delta=1$, we finally get \begin{equation*}\mathbb{L}_{s}\leq\frac{C}{N^{s/2-1}}.\end{equation*}By Theorem \ref{THM_Berry-Essen}, the result follows. \rule{5.0pt}{5.0pt} \par\begin{remark}Applying similar reasoning we estimate the relative entropy distance. For $O\in H_{N}$, \@@amsalign D(\Lambda_{O}\!\parallel\!\mathcal{N})\leq\begin{cases}C\,N^{-1}&\text{for $\delta=1,$}\\ C\,v_{N}^{-2\min\{(\delta-1),\frac{1}{2}\}}&\text{for $\delta>1$},\end{cases} for some $C>0$ and all $N$. \end{remark} \par\par\@@numbered@section{section}{toc}{An example} \par As an example we consider the space $X=\mathbb{Q}_{p}$ equipped with its standard ultrametric $\left|x-y\right|_{p}$ and its normalized Haar measure. Let $\mathcal{D}^{\alpha}$, $\alpha>0$, be a homogeneous hierarchical Laplacian uniquely defined by its eigenvalues \begin{equation*}\lambda{{}^{\alpha}}(B)=\left(\frac{p}{\mathrm{diam}(B)}\right)^{\alpha},\text{ \ }B\in\mathcal{B}.\end{equation*}Let $\mathcal{D}^{\alpha}\left(\omega\right)$ be its random perturbation by i.i.d. $\{\varepsilon(B)\}_{B\in\mathcal{B}}$ as defined in Section \ref{Prem_sec}. As in the previous section we assume that $\varepsilon(B)$ admits a bounded density. We notice that $\mathcal{D}^{\alpha}$ satisfies condition (\ref{lambda-B condition}) with $\delta=2\alpha.$ In particular, for any $\alpha\geq 1/2$ the normalized arithmetic means $\Lambda_{O}^{\alpha}\left(\omega\right)$ converge as $O\rightarrow\varpi$ to the standard normal random variable $Z$ in the sense of the total variation distance. In this section we study convergence assuming that $0<\alpha<1/2.$ \par\begin{theorem}For any $0<\alpha<1/2$ there exists a non-gaussian random variable $\Lambda^{\alpha}$ such that the normalized arithmetic means $\Lambda_{O}^{\alpha}$ converge to $\Lambda^{\alpha}$ in the sense of the total variation distance. \end{theorem} \par\noindent{Proof.} Following line-by-line the proof of the Theorem \ref{Thm_approx} we write \begin{equation*}\sigma\left(\overline{U}_{N}\right)^{2}=\sigma\left(\mathcal{U}_{N}\right)^{2}+\sigma\left(X_{N}\right)^{2}.\end{equation*}As $\lambda_{k}=p^{-\alpha(k-1)}$ and $c_{k}=\left(p^{\alpha}-1\right)p^{-\alpha k}$, we get \begin{equation*}a_{k}=c_{k}/\lambda_{0}=\left(p^{\alpha}-1\right)p^{-\alpha(k+1)},\text{ }k\geq 0.\end{equation*}Using the above data and setting $\sigma^{2}=\mathrm{Var}\left(\varepsilon\right)$ we estimate $\sigma\left(\mathcal{U}_{N}\right)$ and $\sigma\left(X_{N}\right)$ at $\infty$ as follows \@@eqnarray\sigma\left(\mathcal{U}_{N}\right)^{2}&=&\sigma^{2}\left(\frac{a_{0}^{2}}{p^{N}}+\frac{a_{1}^{2}}{p^{N-1}}+...+a_{N}^{2}\right)=\sigma^{2}\left(p^{\alpha}-1\right)^{2}\sum\limits_{0\leq l\leq N}p^{-N+l-2\alpha(l+1)}\\ &\sim&\frac{\sigma^{2}\left(p^{\alpha}-1\right)^{2}}{1-p^{2\alpha-1}}p^{-2\alpha\left(N-1\right)}=\frac{\sigma^{2}}{1-p^{2\alpha-1}}a_{N}^{2}\cr and\@@eqnarray\sigma\left(X_{N}\right)^{2}&=&\sigma^{2}\left(a_{N+1}^{2}+a_{N+2}^{2}+...\right)=\sigma^{2}\left(p^{\alpha}-1\right)^{2}\sum\limits_{l\geq N+1}p^{-2\alpha(l+1)}\\ &\sim&\frac{\sigma^{2}\left(p^{\alpha}-1\right)^{2}}{1-p^{-2\alpha}}p^{-2\alpha(N+2)}=\frac{\sigma^{2}}{1-p^{-2\alpha}}a_{N+1}^{2}.\cr Let $\{\varepsilon_{i}\}_{i\geq 0}$ be i.i.d. random variables independent of $\{\varepsilon(B)\}_{B\in\mathcal{B}}$ and having the same common distribution as $\{\varepsilon(B)\}_{B\in\mathcal{B}}.$ By (\ref{V-L equation}) and (\ref{ass-sigma-V}) the random variable $X_{N}/\sigma\left(X_{N}\right)$ converges in law to the random variable\begin{equation*}X\mathfrak{=}\sqrt{1-p^{-2\alpha}}\left(\frac{\varepsilon_{0}}{\sigma\left(\varepsilon_{0}\right)}+p^{-\alpha}\frac{\varepsilon_{1}}{\sigma\left(\varepsilon_{2}\right)}+...+p^{-k\alpha}\frac{\varepsilon_{k}}{\sigma\left(\varepsilon_{k}\right)}+...\right).\end{equation*}\par Let $\{\varepsilon_{ij}\}_{i,j\geq 0}$ be i.i.d. random variables independent of both $\{\varepsilon_{i}\}_{i\geq 0}$ and $\{\varepsilon(B)\}_{B\in\mathcal{B}}$ and having the same common distribution as $\{\varepsilon(B)\}_{B\in\mathcal{B}}.$ Define the random variables\begin{equation*}S_{k}=\sum\limits_{0\leq j\leq p^{k}}\varepsilon_{kj},\text{ }k=0,1,2,...\text{ .}\end{equation*}By (\ref{U-L equation}) and (\ref{ass-sigma-U}) the random variable $\mathcal{U}_{N}/\sigma\left(\mathcal{U}_{N}\right)$ converges in law to the random variable\begin{equation*}U=\sqrt{1-p^{2\alpha-1}}\sum\limits_{k\geq 0}p^{\left(2\alpha-1\right)k/2}\frac{S_{k}}{\sigma\left(S_{k}\right)}.\end{equation*}Finally, the random variable\begin{equation*}\Lambda_{N}=\frac{\overline{U}_{N}}{\sigma\left(\overline{U}_{N}\right)}=\frac{\sigma\left(\mathcal{U}_{N}\right)}{\sigma\left(\overline{U}_{N}\right)}\frac{\mathcal{U}_{N}}{\sigma\left(\mathcal{U}_{N}\right)}+\frac{\sigma\left(X_{N}\right)}{\sigma\left(\overline{U}_{N}\right)}\frac{X_{N}}{\sigma\left(X_{N}\right)}\end{equation*}converges in law to the random variable\begin{equation*}\Lambda=\sqrt{\frac{1-p^{-2\alpha}}{1-p^{-1}}}U+\sqrt{\frac{p^{-2\alpha}-p^{-1}}{1-p^{-1}}}V\mathfrak{.}\end{equation*}By Cram\'{e}r's theorem $U$ and $V$, and therefore $\Lambda$ are not Gaussian. \par We claim that $\Lambda_{N}$ converges to $\Lambda$ in the total variation distance. To prove the claim we work with characteristic functions. As before for a random variable $X$ we denote by $\Phi_{X}(\xi)=\mathbb{E(}\exp(i\xi X)\mathbb{)}$ its characteristic function. The following inequality holds \@@amsalign\Big{|}\Phi_{\Lambda_{N}}\left(\frac{\sigma\left(\overline{U}_{N}\right)}{a_{N+1}}\xi\right)\Big{|}\leq\big{|}\Phi_{\varepsilon}\left(\xi\right)\big{|}\big{|}\Phi_{\varepsilon}\left(p^{-\alpha}\xi\right)\big{|}. Equations \eqref{ass-sigma-U} and \eqref{ass-sigma-V} yield \@@amsalign A_{N}=\frac{\sigma\left(\overline{U}_{N}\right)}{a_{N+1}}\to\left(\frac{\sigma^{2}(1-p^{-1})}{(p^{-2\alpha}-p^{-1})(1-p^{-2\alpha})}\right)^{1/2}:=A. Moreover $\Phi_{\Lambda_{N}}(A_{N}\xi)\to\Phi_{\Lambda}(A\xi)$ pointwise. As $\varepsilon$ has a bounded density, $\Phi_{\varepsilon}\in L^{2}(\mathbb{R})$ and therefore the function in the right-hand side of \eqref{ineq111} is in $L^{1}(\mathbb{R})$. This in turn implies that $\Phi_{\Lambda_{N}}(\xi)\to\Phi_{\Lambda}(\xi)$ in $L^{1}(\mathbb{R})$. It follows that the density $p_{N}$ of $\Lambda_{N}$ converges pointwise to the density $p_{\Lambda}$ of $\Lambda$. Finally Scheff\'{e}'s lemma \cite[cite]{[\@@bibref{}{VanDerVart}{}{}, Section 2.9]} yields that $\Lambda_{N}\to\Lambda$ in the total variation distance, as desired. \rule{5.0pt}{5.0pt} \par\par\@@unnumbered@section{section}{}{Acknowledgment} We wish to thank G.P. Chistyakov and F. G\"{o}tze for fruitful discussions. \par\thebibliography\lx@bibitem{AizMol} M. Aizenman and S.A. Molchanov, {Localization at Large Disorder and at extreme Energies: An Elementary Derivation}, Communications in Mathematical Physics 157(1993), 245--278. \par\lx@bibitem{AlbeverioKarwowski} S. Albeverio and W. Karwowski, {A random walk on $p$-adic numbers: generator and its spectrum}, Stochastic processes and their Applications 53(1994), 1--22. \par\lx@bibitem{BCW} A. Bendikov, W. Cygan and W. Woess, {Oscillating heat kernels on ultrametric spaces}, arXiv 1610.03292, to appear in Journal of Spectral Theory (2016). \par\lx@bibitem{BGMS} A. Bendikov, A. Grigoryan, S. Molchanov and G. Samorodnitsky, {On a class of random perturbations of the hierarchical Laplacian}, Izvestiya Mathematics 79(2015), 859--893. \par\lx@bibitem{BGP} A. Bendikov, A. Grigoryan and C. Pittet, {On a class of Markov semigroups on discrete ultrametric spaces}, Potential Analysis 37(2012), 125--169. \par\lx@bibitem{BGPW} A. Bendikov, A. Grigoryan, C. Pittet and W. Woess, {Isotropic Markov semigroups on ultrametric spaces}, Russian Mathematical Surveys 69(2014), 589--680. \par\lx@bibitem{BendikovKrupski} A. Bendikov and P. Krupski, {On the spectrum of the hierarchical Laplacian}, Potential Analysis 41(2014), 1247–-1266. \par\lx@bibitem{Bobkov} S.G. Bobkov, G.P. Chistyakov and F. G\"{o}tze, {Berry- Esseen bounds in the entropic central limit theorem}, Probability Theory and Related Fields 159(2014), 435--478. \par\lx@bibitem{Figa-Tal1} M. Del Muto and A. Fig\`{a}-Talamanca, {Diffusion on locally compact ultrametric spaces}, Expositiones Mathematicae 22(2004), 197--211. \par\lx@bibitem{Figa-Tal2} M. Del Muto and A. Fig\`{a}-Talamanca,{ Anisotropic diffusion on totally disconnected abelian groups}, Pacific Journal of Mathematics 225(2006), 221--229. \par\lx@bibitem{Dyson2} F.J. Dyson, {Existence of a phase-transition in a one-dimensional Ising ferromagnet}, Communications in Mathematical Physics 12(1969), 91--107. \par\lx@bibitem{Johnson} O. Johnson, {Information Theory and the Central Limit Theorem}, Imperial College Press, 2004. \par\lx@bibitem{Kochubey20004} A.N. 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Pinsker, {Information and information stability of random variables and processes}, Holden-Day, 1964. \par\lx@bibitem{Shiryaev} A.N. Shiryaev, {Probability}, Second edition, Springer, 1996. \lx@bibitem{Taibleson75} M.H. Taibleson, {Fourier analysis on local fields}, Princeton University Press, 1975. \par\lx@bibitem{VanDerVart} A.W. van der Vaart, {Asymptotic statistics}, Cambridge University Press, 1998. \par\lx@bibitem{Vladimirov} V.S. Vladimirov, {Generalized functions over the field of $p$-adic numbers}, Uspekhi Matematicheskikh Nauk 43(1988), 17--53. \par\lx@bibitem{Vladimirov94} V.S. Vladimirov, I.V. Volovich and E.I. Zelenov, {$p$-adic analysis and mathematical physics}, Series on Soviet and East European Mathematics, vol. 1, World Scientific Publishing Co., Inc., River Edge, 1994. \endthebibliography \par\vskip 12.0pt plus 4.0pt minus 4.0pt\par\begin{tabular}[]{ll}{Alexander Bendikov}{}&{Wojciech Cygan}\\ Institute of Mathematics&Institute of Mathematics\\ University of Wroclaw, Poland&University of Wroclaw, Poland\\ bendikov@math.uni.wroc.pl&wojciech.cygan@uwr.edu.pl\\ &\end{tabular} \par\@add@PDF@RDFa@triples\par\end{document}$$

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Mathematics Subject Classification: 12H25, 60F05, 94A17, 47S10, 60J25.

Key words: ultrametric space, $p$-adic numbers, hierarchical Laplacian, fractional derivative, total variation and entropy distance.

1 Introduction

The concept of hierarchical lattice and hierarchical distance was proposed by F.J. Dyson in his famous paper on the phase transition for $\mathbf{1D}$ ferromagnetic model with long range interaction [11]. The notion of the hierarchical Laplacian $L$, which is closely related to the Dyson’s model was studied in several mathematical papers [14], [15], [16], [17], [2], [6], [7] and [3]. These papers contain some basic information about $L$ (the spectrum, the Markov semigroup, resolvent etc). In the case when the state space is discrete and the hierarchical lattice satisfies some symmetry conditions (homogenuity, self-similarity etc) it can be identified with some discrete infinitely generated Abelian group $G$ equipped with a translation invariant ultrametric $d$ and with a Haar measure $m$. The Markov semigroup $P^{t}=\exp(-tL)$ acting on $L^{2}(G,m)$ becomes then symmetric, translation invariant and isotropic. In particular, $\mathsf{Spec}(L)$ is pure point and all eigenvalues have infinite multiplicity.

In paper [4] we study a class of random perturbations of hierarchical Laplacians $L.$ Each outcome $L^{\omega},\omega\in\Omega,$ of the perturbed hierarchical Laplacian is by itself a hierarchical Laplacian whence its spectrum $\mathsf{Spec}(L^{\omega})$ is still pure point (with compactly supported eigenfunctions). Using the classical average procedure one defines the integrated density of states. Contrary to the deterministic case it may admit a continuous density w.r.t. $m$, the density of states. The density of states detects the spectral bifurcation from the pure point spectrum to the continuous one. The eigenvalues form locally a Poisson point process with intensity given by the density of states. The normalized sequence of arithmetic means of $L^{\omega}$-eigenvalues converges in law to the standard normal distribution. In this note we study the convergence in relative entropy, in particular in the total variation distance. Under certain mild assumptions we establish the rate of convergence.

2 Preliminaries

Hierarchical lattice.

Let $(X,d)$ be a proper non-compact ultrametric space. Recall that proper metric space means that all closed balls are compact, and ultrametric $d$ is a metric which is an ultrametric, that is

[TABLE]

A basic consequence is that any two balls are either disjoint or one is contained in the other. The collection of all balls with a fixed positive radius forms a countable partition of $X$, and decreasing the radius leads to a refined partition. This is consistent with the structure of “Hierarchical lattice” as in the old papers, going back to [11].

Let $m$ be a Radon measure on $X$ such that $m(X)=\infty$ and $m(B)>0$ for each closed ball which is not a singleton, and $m(\{a\})>0$ if and only if $a$ is an isolated point of $X$. Let $\mathcal{B}$ be the collection of all balls with $m(B)>0$. Each $B\in\mathcal{B}$ has a unique predecessor or parent $B^{\prime}\in\mathcal{B}\setminus\{B\}$ which contains $B$ and is such that $B\subseteq D\subseteq B^{\prime}$ for $D\in\mathcal{B}$ implies $D\in\{B,B^{\prime}\}$. In this case, $B$ is called a successor of $B^{\prime}$. Since $X$ is proper, each non-singleton ball has only finitely many (and at least 2) successors. Their number is the degree of the ball.

Hierarchical Laplacian.

We consider a function $C:\mathcal{B}\rightarrow(0\,,\,\infty)$ which satisfies, for all $B\in\mathcal{B}$ and all non-isolated $a\in X$,

[TABLE]

Let $\mathcal{F}$ be the set of all locally constant functions having compact support. It is known that $\mathcal{F}$ consists of continuous functions and is dense in all $L^{p}(X,m).$ Given the space $X$, the measure $m$ and the function $C:\mathcal{B}\rightarrow(0\,,\,\infty)$, we define (pointwise) the hierarchical Laplacian $L_{C}\,$: for each $f$ in $\mathcal{F}$ and $x\in X$ we set

[TABLE]

The operator $(L_{C},\mathcal{F})$ acts in $L^{2}(X,m)$, is symmetric and admits a complete system of eigenfunctions $\{f_{B}:B\in\mathcal{B}\}$ given by

[TABLE]

The eigenvalue corresponding to $f_{B}$ depends only on $B^{\prime}$ and is $\lambda(B^{\prime})$, as given in (2.1). Since all $f_{B}$ belong to $\mathcal{F}$ and the system $\{f_{B}:B\in\mathcal{B}\}$ is complete we conclude that $(L_{C},\mathcal{F})$ is an essentially self-adjoint operator. By a slight abuse of notation, we shall write $(L_{C},\mathsf{Dom}_{L_{C}})$ for its unique self-adjoint extension. For all of this we refer to [7], [5] and [6].

Homogeneous hierarchical Laplacian.

For the analysis undertaken in this paper, we require that the ultrametric measure space $(X,d,m)$ and the hierarchical Laplacian $L_{C}$ are homogeneous, that is there exists a group of isometries of $(X,d)$ which

• •

acts transitively on $X$, and

• •

leaves both the reference measure $m$ and the function $C(B)$ invariant.

The first assumption implies that $(X,d)$ is either discrete or perfect. Basic examples which we have in mind are

$X=\mathbb{Q}_{p}$ – the ring of $p$-adic numbers, where $p\geq 2$ (integer). 2. 2.

$X=\bigoplus_{j=1}^{\infty}\mathbb{Z}/p_{j}\,\mathbb{Z}$ – the direct sum of countably many cyclic groups. 3. 3.

$X=S_{\infty}\,$ – the infinite symmetric group, that is, the group of all permutations of the positive integers that fix all but finitely many elements.

The homogeneity assumptions and the fact that $X$ is non-compact imply that we have the following two cases.

• Case 1.

$\;X$ is perfect, and $\{d(x,y):y\in X\}=\{0\}\cup\{r_{k}:k\in\mathbb{Z}\}$, where $r_{k}<r_{k+1}$ with $\lim\limits_{k\rightarrow\infty}r_{k}=\infty$ and $\lim\limits_{k\rightarrow-\infty}r_{k}=0\,$;

• Case 2.

$\;X$ is countable, and $\{d(x,y):y\in X\}=\{r_{k}:k\in\mathbb{N}_{0}\}$, where $r_{0}=0$, $r_{k}<r_{k+1}$ with $\lim\limits_{k\rightarrow\infty}r_{k}=\infty\,$.

In both cases, we let $\mathcal{B}_{k}$ be the collection of all closed balls of diameter $r_{k}$. This is a partition of $X$, and it is finer than $\mathcal{B}_{k+1}$. By homogeneity, all balls in $\mathcal{B}_{k}$ are isometric. In particular, the number $n_{k}$ of successor balls is the same for each ball in $\mathcal{B}_{k}\,$, where $k\in\mathbb{Z}$ in Case 1, and $k\in\mathbb{N}_{0}$ in Case 2. We notice that the degree sequence $(n_{k})$ satisfies $2\leq n_{k}<\infty$.

It is useful to associate an infinite tree with $X$, see Figure 1. Its vertex set is $\mathcal{B}$, and there is an edge between any $B\in\mathcal{B}$ and its predecessor $B^{\prime}$. In this situation, $\mathcal{B}_{k}$ is the horocyle $H_{k}$ of the tree with index $k$, and $X$ is the (lower) boundary of that tree. For more details see [6], and [9], [10].

For having homogeneity, the reference measure $m$ is also uniquely defined up to a constant factor. If we set $m(B)=1$, for each $B\in\mathcal{B}_{0}$, then, for any $k\in\mathbb{Z}$ (Case 1), resp. $k\in\mathbb{N}_{0}$ (Case 2), and for $B\in\mathcal{B}_{k}$,

[TABLE]

This determines $m$ uniquely as a measure on the Borel $\sigma$-algebra of $X$. Regarding the hierarchical Laplacian, homogeneity means that $C(B)=C_{k}$ is the same for each $B\in\mathcal{B}_{k}\,$. Along with the function $C(B)$, also the eigenvalues of (2.1) depend only on $k$:

[TABLE]

As noticed in [9], [10], the homogeneous ultrametric measure space $(X,m)$ can then be identified with a locally compact totally disconnected group $G$ equipped with its Haar measure. In fact, we may even identify it with an Abelian group. If $(r_{k})$ is the sequence of distances defied above then $G_{k}=B(e,r_{k})$ is a compact-open subgroup of $G\equiv X$,

[TABLE]

gives the degree sequence. The collection $\mathcal{B}_{k}$ of balls with diameter $r_{k}$ consists of the left cosets of $G_{k}$ in $G$. We usually normalize the Haar measure $m$ such that $m(G_{0})=1$.

[TABLE]