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 centered steps attracted to a stable law $Y$ with index $1 < α\leq 2$ and positivity parameter $ρ= P(Y>0)$, we find $λ(μ)\sim Kμ^{(1-ρ)/ν}$, $ν= 1-1/α$. Invisible in the driftless theory, skewness tunes this exponent continuously once a drift is present, through $ρ$ alone, across $[1,\,1/(α-1)]$. The formula recovers the Gaussian linear law with slope $\sqrt{2}$ and, for symmetric heavy tails, the power $μ^{α/2(α-1)}$. It is exact for Gaussian and strictly stable steps and gives the leading power throughout the corresponding domains of attraction, with $K$ explicit for strictly stable steps. The results follow directly from one Mellin transform of the harmonic sum in the Spitzer-Baxter identity, whose poles deliver at once the leading law, its prefactor, and a correction ladder, unifying diffusive, heavy-tailed, and skewed walks. The same transform also yields the expected maximum, recovering Kingman&#39;s heavy-traffic law and Siegmund&#39;s corrected-diffusion constant as adjacent poles.