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

A random walk with small positive drift $\mu$ sets new records at a rate $\lambda(\mu)$ that vanishes as $\mu \to 0$.

For centered steps attracted to a stable law $Y$ with index $1 < \alpha \leq 2$ and positivity parameter $\rho = P(Y>0)$, we find $\lambda(\mu)\sim K\mu^{(1-\rho)/\nu}$, $\nu = 1-1/\alpha$.

Invisible in the driftless theory, skewness tunes this exponent continuously once a drift is present, through $\rho$ alone, across $[1,\,1/(\alpha-1)]$.

The formula recovers the Gaussian linear law with slope $\sqrt{2}$ and, for symmetric heavy tails, the power $\mu^{\alpha/2(\alpha-1)}$.

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's heavy-traffic law and Siegmund's corrected-diffusion constant as adjacent poles.