Random walk and Brownian local times in Wiener sheets: a tribute to my almost surely most visited \(75\) years young best friends, Endre Csáki and Pál Révész
type: Item
instance of: scholarly article
title: Random walk and Brownian local times in Wiener sheets: a tribute to my almost surely most visited \(75\) years young best friends, Endre Csáki and Pál Révész
zbMATH Open document ID: 1240.60058
author: Miklós Csörgo
published in: Periodica Mathematica Hungarica
publication date: 2011-12-19T00:00:00Z
review text: In this tribute to two outstanding probabilists, on the occasion of their 75th birthday, the author focusses on one of their many lasting contributions to modern probability theory, namely, the asymptotic theory of random walk and Brownian local times and their interplay revealed by strong approximations. In Section 1 of his exposition he concentrates on presenting asymptotic properties of random walk local times that can be studied via invariance principles in terms of Brownian local times. To be more specific, let \[ \xi(k,n)=\#\{i:1\leq i\leq n, S_i=k\},\;k=0,\pm1,\pm2,\ldots,\;n=1,2,\ldots, \] be the local time process of the simple symmetric random walk \(S_i=X_1+\cdots+X_i,\;i=1,2,\ldots\), where \(X_i,\;i=1,2,\ldots,\) are i.i.d.\ random variables with distribution \(P(X_i=1)=1/2=P(X_i=-1)\). Correspondingly, let \(\eta(x,t)\) denote the local time of a standard Wiener process \(W=\{W(t),t\geq0\}\), that is, \[ \lambda\{s:0\leq s\leq t, W(s)\in A\}=\int_A\eta(x,t)\,dx\hbox{ a.s.} \] for any Borel set \(A\) on the real line, where \(\lambda\) is the Lebesgue measure. Then, on a rich enough probability space, as \(n\to\infty\), \[ \sup_x| \xi(x,n)-\eta(x,n)| =o(n^{1/4+\varepsilon})\hbox{ a.s.} \] for any \(\varepsilon>0\), where the supremum is taken over all integers \(x\); see [\textit{P. Révész}, Analytical methods in probability theory, Proc.\ Conf., Oberwolfach 1980, Lect.\ Notes Math.\ 861, 128--145 (1981; Zbl 0456.60029)] for this result and [\textit{E. Csáki} and \textit{P. Révész}, Z. Wahrscheinlichkeitstheor.\ Verw.\ Geb.\ 62, 263--278 (1983; Zbl 0488.60045)] for an extension to the case of an integer-valued recurrent random walk. The author puts these approximations into their historical context and discusses the best possible rate of approximation, which cannot go beyond \(O(n^{1/4} (\log\log n)^{3/4})\). In Section 2 the emphasis is on the centered local time processes \(\xi(k,n)-\xi(0,n)\) and \(\eta(x,t)-\eta(0,t)\). Again, the history of the sub…
Mathematics Subject Classification ID: 60F15
Mathematics Subject Classification ID: 60F17
Mathematics Subject Classification ID: 60G15
Mathematics Subject Classification ID: 60G50
Mathematics Subject Classification ID: 60J55
zbMATH DE Number: 5990584
zbMATH Keywords: random walk
zbMATH Keywords: Brownian motion
zbMATH Keywords: strong approximations
zbMATH Keywords: random walk and Brownian local times
zbMATH Keywords: invariance principles
zbMATH Keywords: Wiener sheet
zbMATH Keywords: local times and additive functionals
zbMATH Keywords: iterated processes
zbMATH Keywords: limit theorems
MaRDI profile type: Publication
full work available at URL: s10998-010-3001-7
OpenAlex ID: W2100121079
cites work: Rates of convergence to Brownian local time
cites work: Approximation theorems for independent and weakly dependent random vectors
cites work: On the character of convergence to Brownian local time. I
cites work: Q4279769
cites work: Q4854290
cites work: How often does a Harris recurrent Markov chain recur?
cites work: On the zero \(\sum_1^n\pm 1\)
cites work: Be \(n\) fixed, or go to infinity: difficulties are welcome, and almost surely overcome. A tribute to Endre Csáki
cites work: Joint path properties of two of my most favourite friends in mathematics: a tribute to Endre Csáki and Pál Révész
cites work: On best possible approximations of local time
cites work: On Occupation Times for Markoff Processes
cites work: Q4451252
cites work: Some of my favorite results with Endre Csáki and Pál Révész -- a survey
cites work: We like to walk on the comb
cites work: Probabilistic construction of the solution of some higher order parabolic differential equation
cites work: Q3959169
cites work: The sequence of sums of independent random variables
cites work: Limit theorems for Lévy processes and Poisson point processes and their applications to Brownian excursions
cites work: An iterated logarithm law for local time
cites work: Q3908217
cites work: An approximation of partial sums of independent RV'-s, and the sample DF. I
cites work: Q4220133
cites work: A discrete analogue and elementary derivation of 'Levy's equivalence' for Brownian motion
cites work: Q5600228
cites work: Limit theorems for random walks, birth and death processes, and diffusion processes
cites work: A property of Brownian motion paths
DOI: 10.1007/S10998-010-3001-7
description: scientific article; zbMATH DE number 5990584