Exact Statistics of the Gap and Time Interval Between the First Two Maxima of Random Walks

TL;DR

This paper derives exact statistical properties of the gap and time interval between the two highest positions of a one-dimensional random walk, revealing their asymptotic behaviors and scaling forms for various jump distributions.

Contribution

It provides the first exact computation of joint and marginal distributions of the gap and time interval between the top two maxima in random walks with symmetric jump distributions.

Findings

The gap distribution decays as g^{-1 - or large g.

The time interval distribution has an algebraic tail with exponent depending on or rom 1 to 2.

The joint distribution exhibits a scaling form for large g and l with fixed ratio.

Abstract

We investigate the statistics of the gap, G_n, between the two rightmost positions of a Markovian one-dimensional random walker (RW) after n time steps and of the duration, L_n, which separates the occurrence of these two extremal positions. The distribution of the jumps \eta_i's of the RW, f(\eta), is symmetric and its Fourier transform has the small k behavior 1-\hat{f}(k)\sim| k|^\mu with 0 < \mu \leq 2. We compute the joint probability density function (pdf) P_n(g,l) of G_n and L_n and show that, when n \to \infty, it approaches a limiting pdf p(g,l). The corresponding marginal pdf of the gap, p_{\rm gap}(g), is found to behave like p_{\rm gap}(g) \sim g^{-1 - \mu} for g \gg 1 and 0<\mu < 2. We show that the limiting marginal distribution of L_n, p_{\rm time}(l), has an algebraic tail p_{\rm time}(l) \sim l^{-\gamma(\mu)} for l \gg 1 with \gamma(1<\mu \leq 2) = 1 + 1/\mu, and \gamma(0<\mu<1) = 2. For l, g \gg 1 with fixed l g^{-\mu}, p(g,l) takes the scaling form p(g,l) \sim g^{-1-2\mu} \tilde p_\mu(l g^{-\mu}) where \tilde p_\mu(y) is a (\mu-dependent) scaling function. We also present numerical simulations which verify our analytic results.