Precise Asymptotics for a Random Walker's Maximum

Abstract

We consider a discrete time random walk in one dimension. At each time step the walker jumps by a random distance, independent from step to step, drawn from an arbitrary symmetric density function. We show that the expected positive maximum E[M_n] of the walk up to n steps behaves asymptotically for large n as, E[M_n]/\sigma=\sqrt{2n/\pi}+ \gamma +O(n^{-1/2}), where \sigma^2 is the variance of the step lengths. While the leading \sqrt{n} behavior is universal and easy to derive, the leading correction term turns out to be a nontrivial constant \gamma. For the special case of uniform distribution over [-1,1], Coffmann et. al. recently computed \gamma=-0.516068...by exactly enumerating a lengthy double series. Here we present a closed exact formula for \gamma valid for arbitrary symmetric distributions. We also demonstrate how \gamma appears in the thermodynamic limit as the leading behavior of the difference variable E[M_n]-E[|x_n|] where x_n is the position of the walker after n steps. An application of these results to the equilibrium thermodynamics of a Rouse polymer chain is pointed out. We also generalize our results to L\'evy walks.