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

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 ${\mu }_p={\mathrm{\Phi }}^{-1}(p)$, where $p=P(\xi >0)$. In contrast with the symmetric case, we find that for every fixed $p>1\mathrm{/}2$ the mean LIS length grows linearly, $⟨L_n(p)⟩\sim a(p)n$, with $a(p)$ increasing from $0$ at $p=1\mathrm{/}2$ 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 $p=1\mathrm{/}2$, 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 $p=1\mathrm{/}2$ to Gaussian-like for every sampled $p>1\mathrm{/}2$.