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October, 1987 Maximal Increments of Local Time of a Random Walk
Naresh C. Jain, William E. Pruitt
Ann. Probab. 15(4): 1461-1490 (October, 1987). DOI: 10.1214/aop/1176991987

## Abstract

Let $(S_j)$ be a lattice random walk, i.e., $S_j = X_1 + \cdots + X_j$, where $X_1, X_2,\ldots$ are independent random variables with values in $\mathbb{Z}$ and common nondegenerate distribution $F$. Let $\{t_n\}$ be a nondecreasing sequence of positive integers, $t_n \leq n$, and $L^\ast_n = \max_{0\leq j\leq n-t_n}(L_{j+t_n} - L_j)$, where $L_n = \sum^n_{j=1}1_{\{0\}}(S_j)$, the number of times zero is visited by the random walk by time $n$. Assuming that the random walk is recurrent and satisfies a more general condition than being in the domain of attraction of a stable law of index $\alpha > 1$, the following results are obtained: (i) Constants $\beta_n$ are defined such that $\lim \sup L^\ast_n\beta^{-1}_n = 1$ a.s. (ii) If $\lim \sup nt^{-1}_n = \infty$, then constants $\gamma_n$ are defined such that $\lim \inf L^\ast_n\gamma^{-1}_n = 1$ a.s. If $\lim \sup nt^{-1}_n < \infty$, then $\lim \inf(L^\ast_n/\gamma'_n) = 0$ or $\infty$ for any choice of $\gamma'_n$ and a simple test is given to determine which is the case. (iii) If $\lim \log(nt^{-1}_n)/\log_2n = \infty$, then $\beta_n \sim \gamma_n$ and $\lim L^\ast_n\beta^{-1}_n = 1$ a.s. Also, the normalizers are found more explicitly in the domain of attraction case.

## Citation

Naresh C. Jain. William E. Pruitt. "Maximal Increments of Local Time of a Random Walk." Ann. Probab. 15 (4) 1461 - 1490, October, 1987. https://doi.org/10.1214/aop/1176991987

## Information

Published: October, 1987
First available in Project Euclid: 19 April 2007

zbMATH: 0639.60077
MathSciNet: MR905342
Digital Object Identifier: 10.1214/aop/1176991987

Subjects:
Primary: 60J15
Secondary: 60J55

Keywords: $\lim \inf$ behavior , $\lim \sup$ behavior , increments of local time , lattice random walk