Maximal Increments of Local Time of a Random Walk

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.