The Range of a Random Walk in Two-Dimensional Time

Abstract

Let $\lbrack X_{ij}: i > 0, j > 0 \rbrack$ be a double sequence of i.i.d. random variables taking values in the $d$-dimensional lattice $E_d$. Also let $S_{mn} = \sum^m_{i=1} \sum^n_{j=1} X_{ij}$. Then the range of random walk $\lbrack S_{mn}: m > 0, n > 0 \rbrack$ up to time $(m, n)$, denoted by $R_{mn}$, is the cardinality of the set $\lbrack S_{pq}: 0 < p \leqq m, 0 < q \leqq n \rbrack$, i.e., the number of distinct points visited by the random walk up to time $(m,n)$. In this paper a strong law for $R_{mn}$, when $d \geqq 3$, has been established. Namely, it has been proved that $\lim R_{mn}/ER_{mn} = 1$ a.s. as either $(m, n)$ or $m (n)$ tends to infinity.