The m-th Longest Runs of Multivariate Random Sequences

TL;DR

This paper derives exact formulas for the distributions of the m-th longest runs in multivariate random sequences, considering different run length orderings and providing a versatile method applicable to various run-related distributions.

Contribution

It introduces a two-step method to derive exact distributions of the m-th longest runs in multivariate sequences, accommodating different run length definitions.

Findings

Exact formulas for m-th longest runs under two definitions.

Method applicable to joint distributions of multiple run lengths.

Framework adaptable to other run-related distributions.

Abstract

The distributions of the $m$-th longest runs of multivariate random sequences are considered. For random sequences made up of $k$ kinds of letters, the lengths of the runs are sorted in two ways to give two definitions of run length ordering. In one definition, the lengths of the runs are sorted separately for each letter type. In the second definition, the lengths of all the runs are sorted together. Exact formulas are developed for the distributions of the m-th longest runs for both definitions. The derivations are based on a two-step method that is applicable to various other runs-related distributions, such as joint distributions of several letter types and multiple run lengths of a single letter type.