Skewness tunes the small-drift record rate of random walks and Lévy flights

Abstract

A random walk with small positive drift $μ$ sets new records at a rate $λ(μ)$ that vanishes as $μ\to 0$. For Gaussian and strictly stable centered steps whose stable law $Y$ has index $1 < α\leq 2$ and positivity parameter $ρ= \mathbb{P}(Y>0)$, we find $λ(μ) \sim Kμ^{(1-ρ)/ν}$ as $μ\to 0$, where $ν=1-1/α$ and $K$ is explicit. Throughout their domains of attraction, the exponent persists, with a slowly varying factor replacing the constant $K$. The exponent is set by the asymmetry only through $ρ$, sweeping the interval $[1,\,1/(α-1)]$ as the skewness varies. For centered strictly stable steps, $ρ$ also governs the driftless record growth, $\langle R_{N}\rangle \sim N^ρ/Γ(1+ρ)$, which the small-drift law meets at the crossover where the drift takes over. The formula recovers the Gaussian linear law $λ(μ) \sim \sqrt{2}μ/σ$ and, for symmetric heavy tails, the power $μ^{α/2(α-1)}$. It follows from one Mellin transform of the harmonic sum in the Spitzer--Baxter identity, which factorizes into a kernel transform carrying the step distribution and a Riemann zeta function carrying the harmonic weights. Its poles deliver the leading law, its prefactor, and a correction ladder reproducing the known Gaussian and stable series, unifying diffusive, heavy-tailed, and skewed walks. The power law ends at the Cauchy point $α=1$, where $μ$ is a pure location shift: for the strictly Cauchy family the limiting rate vanishes and records accumulate sublinearly with a shift-dependent exponent.