The range of two-parameter random walk in space

Abstract

Let {X ij; i>0, j>0} be a double sequence of i.i.d. random variables taking values in the d-dimensional integer lattice E d . Also let $$S_{mn} = \sum\limits_{i = 1}^m {\sum\limits_{j = 1}^n {X_{ij} } } $$ . Then the range of random walk {S mn: m>0, n>0} up to time (m, n), denoted by R mn , is the cardinality of the set {S pq: 0<p≦m, 0<q≦n}, i.e., the number of distinct points visited by the random walk up to time (m, n). In this paper a sufficient condition in terms of the characteristic function of X 11 is given so that $$\lim \frac{{R_{mn} }}{{ER_{mn} }} = 1$$ a.s. as either (m, n) or m(n) tends to infinity.