Records, drift, and the longest increasing subsequence of biased Gaussian random walks (opens in new tab)

The longest increasing subsequence (LIS) of a random walk has so far 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 $p = \mathbb{P}(\xi>0)$. 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)...