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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 

 

Erickson's conjecture on the rate of escape of $ d$-dimensional random walk


Author: Harry Kesten
Journal: Trans. Amer. Math. Soc. 240 (1978), 65-113
MSC: Primary 60J15; Secondary 60F15
DOI: https://doi.org/10.1090/S0002-9947-1978-0489585-X
MathSciNet review: 489585
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Abstract: We prove a strengthened form of a conjecture of Erickson to the effect that any genuinely d-dimensional random walk $ {S_n},d \geqslant 3$, goes to infinity at least as fast as a simple random walk or Brownian motion in dimension d. More precisely, if $ S_n^\ast$ is a simple random walk and $ {B_t}$, a Brownian motion in dimension d, and $ \psi :[1,\infty ) \to (0,\infty )$ a function for which $ {t^{ - 1/2}}\psi (t) \downarrow 0$, then $ \psi {(n)^{ - 1}}\vert S_n^\ast\vert \to \infty $ w.p.l, or equivalently, $ \psi {(t)^{ - 1}}\vert{B_t}\vert \to \infty $ w.p.l, iff $ \smallint _1^\infty \psi {(t)^{d - 2}}{t^{ - d/2}} < \infty $; if this is the case, then also $ \psi {(n)^{ - 1}}\vert{S_n}\vert \to \infty $ w.p.l for any random walk Sn of dimension d.


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DOI: https://doi.org/10.1090/S0002-9947-1978-0489585-X
Keywords: Random walk, escape rate, concentration functions
Article copyright: © Copyright 1978 American Mathematical Society