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 $λ(μ) \sim Kμ^{(1-ρ)/ν}$, $ν=1-1/α$, as $μ\to 0$. The result is exact for Gaussian and strictly stable steps, and extends at the leading-power level to their domains of attraction. The exponent is set by the asymmetry only through $ρ$, sweeping the interval $[1,\,1/(α-1)]$ as the skewness varies. It recovers the Gaussian linear law with slope $\sqrt{2}$ and, for symmetric heavy tails, the power $μ^{α/2(α-1)}$; beyond the stable tail ratio, distributional details enter through the prefactor $K$, which is explicit for strictly stable steps. The result follows directly from one Mellin transform of the harmonic sum in the Spitzer-Baxter identity, which factorizes into a kernel transform and a Riemann $ζ$ factor 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's heavy-traffic law for queues and Siegmund's corrected-diffusion constant as adjacent poles.