Exact statistics of record increments of random walks and L\'evy flights

TL;DR

This paper derives exact and universal statistics for record increments in random walks and Le9vy flights, revealing stationary increment distributions and a universal decay law for the probability of decreasing record increments.

Contribution

It provides explicit formulas for record increment distributions and proves the universality of the decay of the probability that record increments decrease monotonically.

Findings

Record increment distribution becomes stationary for large n.

Probability Q(n) decays as rac{A}{\u221a n} with universal amplitude.

Q(n) is independent of jump distribution for large n.

Abstract

We study the statistics of increments in record values in a time series $\{x_0=0,x_1, x_2, \ldots, x_n\}$ generated by the positions of a random walk (discrete time, continuous space) of duration $n$ steps. For arbitrary jump length distribution, including L\'evy flights, we show that the distribution of the record increment becomes stationary, i.e., independent of $n$ for large $n$, and compute it explicitly for a wide class of jump distributions. In addition, we compute exactly the probability $Q(n)$ that the record increments decrease monotonically up to step $n$. Remarkably, $Q(n)$ is universal (i..e., independent of the jump distribution) for each $n$, decaying as $Q(n) \sim {\cal A}/\sqrt{n}$ for large $n$, with a universal amplitude ${\cal A} = e/\sqrt{\pi} = 1.53362\ldots$.