Exact statistics of record increments of random walks and Lévy flights

Abstract

We study the statistics of increments in record values in a time series $\{x_0=0,x_1,x_2,\dots ,x_n\}$ generated by the positions of a random walk (discrete time, continuous space) of duration $n$ steps. For arbitrary jump length distribution, including Lévy 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 $Q(n)\sim \mathcal{A}/\sqrt{n}$ for large $n$, with a universal amplitude $\mathcal{A}=e/\sqrt{\pi }=1.53362\dots$.

6 pages + 5 pages of supplemental material, 5 figures. Published version

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• Universal survival probability for a correlated random walk and applications to records
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• Statistical properties of avalanches via the c-record process
• Optimizing Leapover Lengths of Lévy Flights with Resetting
• Optimal conditions for first passage of jump processes with resetting
• Empirical scaling of the length of the longest increasing subsequences of random walks
• On the first positive position of a random walker
• New insights into the distribution of the topmost gap in random walks and Lévy flights