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October, 1980 Further Limit Theorems for the Range of a Two-Parameter Random Walk in Space
Nasrollah Etemadi
Ann. Probab. 8(5): 917-927 (October, 1980). DOI: 10.1214/aop/1176994621


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 $$\lim_{(m,n)\rightarrow\infty}\frac{mn - R_{mn}}{m + n} = \sum^\infty_{l=1}(1 - r^{(l)}) < \infty\quad \mathrm{a.s.}$$


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Nasrollah Etemadi. "Further Limit Theorems for the Range of a Two-Parameter Random Walk in Space." Ann. Probab. 8 (5) 917 - 927, October, 1980.


Published: October, 1980
First available in Project Euclid: 19 April 2007

zbMATH: 0441.60028
MathSciNet: MR586776
Digital Object Identifier: 10.1214/aop/1176994621

Primary: 60F16
Secondary: 60G50 , 60J15

Keywords: genuine dimension , Random walk

Rights: Copyright © 1980 Institute of Mathematical Statistics


Vol.8 • No. 5 • October, 1980
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