Remote Access Transactions of the American Mathematical Society
Green Open Access

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
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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.


References [Enhancements On Off] (What's this?)


Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 60J15, 60F15

Retrieve articles in all journals with MSC: 60J15, 60F15


Additional Information

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