Records, drift, and the longest increasing subsequence of biased Gaussian random walks

Abstract

The longest increasing subsequence (LIS) of a random walk has been studied mainly for zero-mean, symmetric step increments. We numerically investigate the LIS of biased Gaussian random walks, with unit-variance increments and positive drift $\mu_{p} = \Phi^{-1}(p)$, where $p = P(\xi>0)$. In contrast with the symmetric case, we find that for every fixed $p>1/2$ the mean LIS length grows linearly, $\langle L_{n}(p)\rangle \sim a(p)n$, with $a(p)$ increasing from $0$ at $p=1/2$ to $1$ as $p \to 1$. The record count is also linear, with coefficient $\lambda(p)$ fixed by Spitzer's formula for the ascending ladder epoch, and the LIS becomes increasingly aligned with this record skeleton as $p$ grows. At the symmetric point $p=1/2$, the record skeleton collapses to the Sparre Andersen $\sqrt{n}$ scale, while the LIS returns to the finite-variance $\sqrt{n}\log{n}$ regime. Near this limit the record rate has the closed-form small-drift slope $\lambda(\mu_{p}) \simeq \sqrt{2}\,\mu_{p}$, whereas the excess $a(\mu_{p})-\lambda(\mu_{p})$ vanishes more slowly than linearly in the drift, although our data do not resolve a single power law. The empirical distribution of $L_{n}$ also changes across this point, from lognormal-like at $p=1/2$ to Gaussian-like for every sampled $p>1/2$.