Further Limit Theorems for the Range of a Two-Parameter Random Walk in Space

Abstract

Let $\{X_{ij}: i\geqslant 1, j \geqslant 1\}$ be a double sequence of i.i.d. random variables taking values in the $d$-dimensional lattice $E_d$. Also let $S_{kl} = \Sigma^k_{i=1}\Sigma^k_{i=1}\Sigma^l_{j=1}X_{ij}$. Then the range of random walk $\{S_{kl}: k \geqslant 1, l \geqslant 1\}$ up to time $(m, n)$, denoted by $R_{mn}$, is the cardinality of the set $\{S_{kl}: 1 \leqslant k \leqslant m, 1 \leqslant l \leqslant n\}$, i.e., the number of distinct points visited by the random walk up to time $(m, n)$. Let $r^{(l)}$ be the probability that the random walk never hits the origin on the time set $\{(i, l): i \geqslant 1\}$. In this paper a sufficient condition in terms of the characteristic function of $X_{11}$ is given so that $$\lim_{(m,n)\rightarrow\infty}\frac{mn - R_{mn}}{m + n} = \sum^\infty_{l=1}(1 - r^{(l)}) < \infty\quad \mathrm{a.s.}$$