Longest increasing subsequences for distributions with atoms, and an inhomogeneous Hammersley process

TL;DR

This paper studies the asymptotic behavior of the longest increasing subsequence length for distributions with atoms, revealing diverse growth rates and providing explicit estimates through coupling with an inhomogeneous Hammersley process.

Contribution

It characterizes the asymptotic order of the longest increasing subsequence for discrete distributions using a variational problem and coupling methods, extending classical results.

Findings

Discrete distributions exhibit a wide range of growth rates for $L_n$.

Explicit estimates are provided for classical discrete distributions.

The asymptotics of $L_n$ can be deduced for any distribution based on tail behavior.

Abstract

A famous result by Hammersley and Versik-Kerov states that the length $L_n$ of the longest increasing subsequence among $n$ iid continuous random variables grows like $2\sqrt{n}$. We investigate here the asymptotic behavior of $L_n$ for distributions with atoms. For purely discrete random variables, we characterize the asymptotic order of $L_n$ through a variational problem and provide explicit estimates for classical distributions. The proofs rely on a coupling with an inhomogeneous version of the discrete-time continuous-space Hammersley process. This reveals that, in contrast to the continuous case, the discrete setting exhibits a wide range of growth rates between $\mathcal{O}(1)$ and $o(\sqrt{n})$, depending on the tail behavior of the distribution. We can then easily deduce the asymptotics of $L_n$ for a completely arbitrary distribution.