We like to walk on the comb
type: Item
instance of: scholarly article
title: We like to walk on the comb
zbMATH Open document ID: 1240.60103
published in: Periodica Mathematica Hungarica
publication date: 2011-12-19T00:00:00Z
review text: The paper presents simpler cases of results obtained by \textit{E. Csáki}, \textit{M. Csörgő}, \textit{A. Földes} and \textit{P. Révész} in the three papers [Electron.\ J.\ Probab.\ 14, 2371--2390, electronic only (2009; Zbl 1190.60020)], [Publ.\ Math.\ 76, No. ~3, 255--270 (2010; Zbl 1224.60191)] and [Stochastic Processes Appl.\ 121, No. ~6, 1290--1314 (2011; Zbl 1223.60025)]. A random walk is considered, \(C(n)=(C_{1}(n),C_{2}(n))\) on \({\mathbb Z}^{2}\), whose nonnull transition probabilities are \(p((x,y),(x,y\pm 1))=1/2\) for \(y\neq 0\), \(p((x,0),(x,\pm 1))=1/4\), \(p((x,0),(x\pm 1,0))=1/4\). The proof of the following main fact is given: for some independent Wiener processes \(W_{1}\), \(W_{2}\), if \(\eta_{2}\) is the local time of \(W_{2}\) then, for every \(\varepsilon >0\), when \(n\rightarrow \infty\), \(n^{-1/4}| C_{1}(n)-W_{1}(\eta_{2}(0,n))| + n^{-1/2}| C_{2}(n)-W_{2}(n) | =O(n^{\varepsilon -1/8})\) holds a.s. The presented results concerning \(W_{1}\), \(W_{2}\) are the following. As \(t\rightarrow \infty\), \(((W_{1}(xt)/b(t),W_{2}(xt)/b(t),\eta_{2}(0,xt)/b(t));0\leq x\leq 1)\), \(b(t)=(2t\log\log t)^{1/2}\), is almost surely relatively compact in the space \(C([0,1],{\mathbb R}^{3})\) and the set of its limit points \((f,g,h)\) is described by: absolutely continuous, null in \(0\), \(\int_{_{0}}^{^{1}}(\dot{f}^{2}(x)+\dot{g}^{2}(x)+\dot{h}^{2}(x))dx\leq 1\), \(g(x)\dot{h}(x)=0\) a.e., nondecreasing \(h\). A similar result with two components, the first \(W_{1}(\eta_{2}(0,xt))/b_{1}(t)\), the second \(W_{2}(xt)/b(t)\) and a set of limit points denoted by \(S^{(2)}\), is shown. For a nonincreasing \(\beta >0\), \(\liminf_{t\rightarrow \infty }sup_{0\leq s\leq t}| u_{j}(\eta_{2}(0,s))| /t^{1/4}\beta (t)=0\) or \(\infty\) when \(\int_{_{1}}^{^{\infty }}(\beta^{j}(t)/t)dt=\infty\), \(<\infty\) respectively, where \(j=1, 2\), \(u_{1}=\eta_{1}(0,\cdot)\), \(u_{2}=W_{1}\). As \(t\rightarrow \infty\), \(P(\sup_{x\in {\mathbb R}}\eta_{1}(x,\eta_{2}(0,t)) >…
Mathematics Subject Classification ID: 60F17
Mathematics Subject Classification ID: 60G50
Mathematics Subject Classification ID: 60J65
Mathematics Subject Classification ID: 60F15
Mathematics Subject Classification ID: 60J10
zbMATH DE Number: 5990589
zbMATH Keywords: random walk
zbMATH Keywords: \(2\)-dimensional comb
zbMATH Keywords: strong approximation
zbMATH Keywords: \(2\)-dimensional Wiener process
zbMATH Keywords: local time
zbMATH Keywords: laws of the iterated logarithm
zbMATH Keywords: iterated Brownian motion
author: Antónia Földes
reviewed by: Ion Cuculescu
MaRDI profile type: Publication
full work available at URL: s10998-010-3165-1
OpenAlex ID: W2033844009
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DOI: 10.1007/S10998-010-3165-1
description: scientific article; zbMATH DE number 5990589