Asymptotic Behavior of the Local Time of a Recurrent Random Walk

Abstract

Let $(S_j)$ be a lattice random walk, i.e. $S_j = X_1 + \cdots + X_j$ where $X_1, X_2, \cdots$ 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.