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 ⩾ 3 {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 ∗ S_n^\ast is a simple random walk and B t {B_t} , a Brownian motion in dimension d, and ψ : [ 1 , ∞ ) → ( 0 , ∞ ) \psi :[1,\infty ) \to (0,\infty ) a function for which t − 1 / 2 ψ ( t ) ↓ 0 {t^{ - 1/2}}\psi (t) \downarrow 0 , then ψ ( n ) − 1 | S n ∗ | → ∞ \psi {(n)^{ - 1}}|S_n^\ast | \to \infty w.p.l, or equivalently, ψ ( t ) − 1 | B t | → ∞ \psi {(t)^{ - 1}}|{B_t}| \to \infty w.p.l, iff ∫ 1 ∞ ψ ( t ) d − 2 t − d / 2 > ∞ \smallint _1^\infty \psi {(t)^{d - 2}}{t^{ - d/2}} > \infty ; if this is the case, then also ψ ( n ) − 1 | S n | → ∞ \psi {(n)^{ - 1}}|{S_n}| \to \infty w.p.l for any random walk Sn of dimension d.