On the rate of approximation in finite-alphabet longest increasing subsequence problems

TL;DR

This paper investigates how quickly the distribution of the longest increasing subsequence length converges to its limiting distribution for finite-alphabet sources, providing specific convergence rates for various cases.

Contribution

It establishes convergence rates of a0 n / c f for finite-alphabet sources and improves the rate to 1/c n for the binary case.

Findings

Convergence rate of a0 n / c f for finite-alphabet sources.

Improved convergence rate of 1/c n for binary sources.

Quantitative analysis of the approximation rate to the limiting distribution.

Abstract

The rate of convergence of the distribution of the length of the longest increasing subsequence, toward the maximal eigenvalue of certain matrix ensembles, is investigated. For finite-alphabet uniform and nonuniform i.i.d. sources, a rate of $\log n/\sqrt{n}$ is obtained. The uniform binary case is further explored, and an improved $1/\sqrt{n}$ rate obtained.