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- W2012330252 abstract "Let ${X_{ij}: igeqslant 1, j geqslant 1}$ 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 $$lim_{(m,n)rightarrowinfty}frac{mn - R_{mn}}{m + n} = sum^infty_{l=1}(1 - r^{(l)}) < inftyquad mathrm{a.s.}$$" @default.
- W2012330252 created "2016-06-24" @default.
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- W2012330252 date "1980-10-01" @default.
- W2012330252 modified "2023-09-27" @default.
- W2012330252 title "Further Limit Theorems for the Range of a Two-Parameter Random Walk in Space" @default.
- W2012330252 doi "https://doi.org/10.1214/aop/1176994621" @default.
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