Abstract: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 $\lambda(\mu) \sim K\mu^{(1-\rho)/\nu}$, $\nu=1-1/\alpha$, as $\mu \to 0$. 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 $\rho$, sweeping the interval $[1,\,1/(\alpha-1)]$ as the skewness varies. It recovers the Gaussian linear law with slope $\sqrt{2}$ and, for symmetric heavy tails, the power $\mu^{\alpha/2(\alpha-1)}$; 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 $\zeta$ 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.
From: J. Ricardo G. Mendonça [view email]
[v1]
Mon, 22 Jun 2026 16:26:06 UTC (114 KB)
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