Skewness tunes the small-drift record rate of random walks and L\'{e}vy flights

Abstract

A random walk with small positive drift $\mu$ sets new records at a rate $\lambda(\mu)$ that vanishes as $\mu \to 0$. For Gaussian and strictly stable centered steps whose stable law $Y$ has index $1 < \alpha \leq 2$ and positivity parameter $\rho = \mathbb{P}(Y>0)$, we find $\lambda(\mu) \sim K\mu^{(1-\rho)/\nu}$ as $\mu \to 0$, where $\nu=1-1/\alpha$ 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 $\rho$, sweeping the interval $[1,\,1/(\alpha-1)]$ as the skewness varies. For centered strictly stable steps, $\rho$ also governs the driftless record growth, $\langle R_{N}\rangle \sim N^{\rho}/\Gamma(1+\rho)$, which the small-drift law meets at the crossover where the drift takes over. The formula recovers the Gaussian linear law $\lambda(\mu) \sim \sqrt{2}\mu/\sigma$ and, for symmetric heavy tails, the power $\mu^{\alpha/2(\alpha-1)}$. 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 $\alpha=1$, where $\mu$ is a pure location shift: for the strictly Cauchy family the limiting rate vanishes and records accumulate sublinearly with a shift-dependent exponent.