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On the number of distinct sites visited by a random walk

✍ Scribed by M. D. Donsker; S. R. S. Varadhan

Publisher: John Wiley and Sons
Year: 1979
Tongue: English
Weight: 727 KB
Volume: 32
Category: Article
ISSN: 0010-3640
DOI: 10.1002/cpa.3160320602

No coin nor oath required. For personal study only.

✦ Synopsis

In the expression for a(?), M is a symmetric measure on the unit ball means that the support of M spans &.

Let {rjl} be the n-step transition probabilities. It follows from hypotheses (i) and (ii) above that for any j e Z d there exists an no such that .rrYu>O. Moreover, from hypothesis (ii) it follows that as n + 00 S d -1 in ' R, (M(dy) = M(d(-y))), and the assumption of non-degeneracy provided n/ct + k as n + 03. In other words, hypotheses (i) and (ii) imply that the random walk is irreducible and that the distribution 7~ belongs to the domain of normal attraction of a non-degenerate symmetric stable law of index a, O < a 12.

Let {x(s), O S s < ~} be a symmetric stable process in Rd of index a, 0 < a 5 2, corresponding to a(t) and let Ln be its infinitesimal generator.

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