On the distribution of random walk local time
type: Item
instance of: scholarly article
title: On the distribution of random walk local time
zbMATH Open document ID: 0608.60065
author: A. N. Borodin
published in: Annales de l'Institut Henri Poincaré. Probabilités et Statistiques
publication date: 1987-01-01T00:00:00Z
full work available at URL: http://www.numdam.org/item
full work available at URL: https://eudml.org/doc/77292
review text: Let \(\{\xi_{\ell}\}^{\infty}_{\ell =1}\) be i.i.d. random variables, \(E\xi_ 1=0\). \(\xi_ 1\) is assumed to have integer values, and \(| E \exp (it\xi_ 1)| =1\) if and only if t is a multiple of \(2\pi\). Let \(\nu_ k=\xi_ 1+...+\xi_ k\), \(1_ A(\cdot)\) be the indicator function of set A, and \(\phi (n,r)=\sum^{n}_{k=1}1_{\{r\}}(\nu_ k)\). The function \(\phi\) (n,r) is the local time of random walk \(\nu_ k\). It is known [the author, Teor. Veroyatn. Primen. 26, 769-783 (1981; Zbl 0474.60056)] that the process \[ {\mathfrak Z}_ n(t,x):=n^{- 1/2}\phi^{([nt],[x\sqrt{n}])} \] converges weakly as \(n\to \infty\) to Brownian local time. Limit theorems of large deviations type are proved for probabilities P(\({\mathfrak Z}_ n(1,x)=y)\) and P(\({\mathfrak Z}_ n(1,x)\geq y)\) when \(\xi_ 1\) satisfies Cramér's condition. Similar results including explicit asymptotic expansions are proved when \(\xi_ 1\) satisfies the condition E \(| \xi_ 1|^{2+m+\delta}<\infty\).
Mathematics Subject Classification ID: 60G50
Mathematics Subject Classification ID: 60J55
Mathematics Subject Classification ID: 60F10
zbMATH DE Number: 3984253
zbMATH Keywords: local time of random walk
zbMATH Keywords: Brownian local time
zbMATH Keywords: Cramér's condition
zbMATH Keywords: asymptotic expansions
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 3984253