On the rate of convergence of the shape of Young diagrams associated with random words

TL;DR

This paper investigates the convergence rates of the shapes of Young diagrams derived from random words, extending previous results beyond uniform distributions and applying to longest increasing subsequences.

Contribution

It provides new rates of convergence in Kolmogorov distance for Young diagram shapes associated with non-uniform random words, including multiple sequences.

Findings

Established convergence rates for Young diagram shapes beyond uniform cases.

Extended results to longest common and increasing subsequences in multiple words.

Quantified the rate of convergence for the length of the longest increasing subsequences.

Abstract

We revisit, beyond the uniform case, some aspects of the convergence of the cumulative shape of the RSK Young diagrams associated with random words, obtaining rates of convergence in Kolmogorov's distance. Since the length of the top row of the diagrams is the length of the longest increasing subsequences of the word, a corresponding rate result follows. This is then extended to the length of the longest common and increasing subsequences in two or more random words.