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We prove a strengthened form of a conjecture of Erickson to the effect that any genuinely d-dimensional random walk S,, d > 3, goes to infinity at least as fast as a simple random walk or Brownian motion in dimension d. More precisely, if S,* is a simple random walk and B, a Brownian motion in dimension d, and J: [1, oo) (O,oo) a function for which t1/24(t)10, then 4(n)'15*1 oo w.p.l, or equivalently, +(t)I1B,I -* w.p. 1, iff f Ip(t)d-2tUd/2 < oo; if this is the case, then also k(n)-l1S,I oo w.p.l for any random walk S,, of dimension d.

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