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

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.