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February, 1984 Asymptotic Behavior of the Local Time of a Recurrent Random Walk
Naresh C. Jain, William E. Pruitt
Ann. Probab. 12(1): 64-85 (February, 1984). DOI: 10.1214/aop/1176993374


Let $(S_j)$ be a lattice random walk, i.e. $S_j = X_1 + \cdots + X_j$ where $X_1, X_2, \cdots$ are independent random variables with values in the integer lattice and common nondegenerate distribution $F$, and let $L_n(x) = \sum^{n - 1}_{j = 0} 1_{\{x\}} (S_j)$, the local time of the random walk at $x$ before time $n$. Define $G(x) = P\{|X_1| > x\}, K(x) = x^{-2} \int_{|y| \leq x}y^2 dF (y), Q(x) = G(x) + K(x)$ for $x > 0. Q$ is continuous, strictly decreasing for large $x$, and tends to zero. Thus $a_y$ may be defined by $Q(a_y) = y^{-1}$ for large $y$ and then we let $c_n = a_{n/\log\log n}$. The basic assumption is that $\lim \sup_{x \rightarrow \infty} G(x)/K(x) < 1$ and $EX_1 = 0$. We prove that there exist positive constants $\theta_1, \theta_2$ such that $\lim \sup_{n \rightarrow \infty} c_nn^{-1}L_n(x) = \theta_1$ a.s. for all $x, \lim \sup_{n \rightarrow \infty} \sup_xc_nn^{-1}L_n(x) = \theta_2$ a.s. Furthermore $\lim_{\delta \rightarrow 0} \lim \sup_{n \rightarrow \infty} \sup_{|x - y| \leq \delta c_n}c_nn^{-1}|L_n(x) - L_n(y)| = 0 a.s.$ One of the main tools is an estimate for the "absolute potential kernel": $\sum^\infty_{n = 0}| P\{S_n = z\} - P\{S_n = z + x\}| \leq C(|x|Q(|x|))^{-1}$ assuming strong aperiodicity.


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Naresh C. Jain. William E. Pruitt. "Asymptotic Behavior of the Local Time of a Recurrent Random Walk." Ann. Probab. 12 (1) 64 - 85, February, 1984.


Published: February, 1984
First available in Project Euclid: 19 April 2007

zbMATH: 0538.60074
MathSciNet: MR723730
Digital Object Identifier: 10.1214/aop/1176993374

Primary: 60J15
Secondary: 60J55

Keywords: absolute potential kernel , lattice random walk , Local time , recurrent

Rights: Copyright © 1984 Institute of Mathematical Statistics


Vol.12 • No. 1 • February, 1984
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