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This paper is devoted to the study of the additive functional t → t 0 f (W (s))ds, where f denotes a measurable function and W is a planar Brownian motion. Kasahara and Kotani  have obtained its second-order asymptotic behaviors, by using the skew-product representation of W and the ergodicity of the angular part. We prove that the vector (· 0 f j (W… (More)
We study the path behaviour of a simple random walk on the 2-dimensional comb lattice 2 that is obtained from 2 by removing all horizontal edges off the x-axis. In particular, we prove a strong approximation result for such a random walk which, in turn, enables us to establish strong limit theorems, like the joint Strassen type law of the iterated logarithm… (More)
We study the occupation measure of various sets for a symmetric transient random walk in Z d with finite variances. Let µ X n (A) denote the occupation time of the set A up to time n. It is shown that sup x∈Z d µ X n (x + A)/ log n tends to a finite limit as n → ∞. The limit is expressed in terms of the largest eigenvalue of a matrix involving the Green's… (More)
Strong theorems are given for the maximal local time on balls and subspaces for the d-dimensional simple symmetric random walk.