Maxima of Two Random Walks: Universal Statistics of Lead Changes

Abstract

International audienceWe investigate statistics of lead changes of the maxima of two discrete-time random walks in one dimension. We show that the average number of lead changes grows as π ^(−1) ln t in the long-time limit. We present theoretical and numerical evidence that this asymptotic behavior is universal. Specifically, this behavior is independent of the jump distribution: the same asymptotic underlies standard Brownian motion and symmetric Lévy flights. We also show that the probability to have at most n lead changes behaves as t^(−1/4) (ln t)^n for Brownian motion and as t ^(−β(µ)) (ln t)^n for symmetric Lévy flights with index µ. The decay exponent β ≡ β(µ) varies continuously with the Lévy index when 0 2