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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
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 $\psi {(n)^{ - 1}}|S_n^\ast | \to \infty$ w.p.l, or equivalently, $\psi {(t)^{ - 1}}|{B_t}| \to \infty$ w.p.l, iff $\smallint _1^\infty \psi {(t)^{d - 2}}{t^{ - d/2}} < \infty$; if this is the case, then also $\psi {(n)^{ - 1}}|{S_n}| \to \infty$ w.p.l for any random walk Sn of dimension d.

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Keywords: Random walk, escape rate, concentration functions
Article copyright: © Copyright 1978 American Mathematical Society