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
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 subject is pointed out, and several important achievements by \textit{E. Csáki} and \textit{P. Révész} are detailed, including laws of the iterated logarithms and strong approximations. Some of them were obtained quite recently. Section 3 explains why iterated processes like \(W(\eta(x,t))\) and their properties are important for the derivation of some of the results in the previous sections.
scientific article; zbMATH DE number 5990584
scientific article