Skewness tunes the small-drift record rate of random walks and Lévy 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 $1<\alpha \le 2$ 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, $⟨R_N⟩\sim N^{\rho }/\Gamma (1+\rho )$, 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.

Preprint, 19 pages, 4 figures, 33 references. In the latest version the results are derived in greater detail