Longest weakly increasing subsequences of discrete random walks on the integers with heavy tailed distribution of increments

TL;DR

This paper studies the length of the longest weakly increasing subsequences in heavy-tailed random walks on integers, revealing different scaling behaviors depending on the tail heaviness of the increment distribution.

Contribution

It provides empirical analysis of how the longest weakly increasing subsequence length scales with n for heavy-tailed increments, extending previous results and proposing a lognormal approximation.

Findings

For finite variance increments, the average length scales as √n log n.

For infinite variance increments, the length scales as n^θ with θ > 0.5.

The distribution of L_n is well-approximated by a lognormal distribution.

Abstract

We investigate the behavior of the length of the longest weakly increasing subsequences (weak LIS) of $n$-step random walks with nonzero integer increments $k = \pm 1, \pm 2, \dots$ given by a zero-mean, symmetric heavy tailed mass distribution proportional to $|k|^{-1-\alpha}$ for several values of the real parameter $\alpha > 0$ together with that of the simple random walk ($k=\pm 1$), to which the $n$-step heavy tailed random walks tends when $\alpha > (1+o(1))\log_{2}{n}$. By means of exploratory fits, weighted nonlinear least squares, and ANOVA model comparison, we found that the sample average length $\langle{L_{n}}\rangle$ scales like $\langle{L_{n}}\rangle \sim \sqrt{n}\log{n}$ when the distribution of increments has finite variance ($\alpha > 2$) and $\langle{L_{n}}\rangle \sim n^{\theta}$ with a varying exponent $\theta > 0.5$ when the variance is infinite ($\alpha \leq 2$). Distributional diagnostics indicate that the bulk of the $L_{n}$ distribution is very well-approximated by a lognormal model, though systematic deviations are observed in the tails. Our results corroborate and expand upon previous results for the LIS of other types of heavy-tailed random walks and raise a conjecture as to whether the distribution of $L_{n}$ is given, or can be effectively described, by a lognormal distribution.