Joint distribution of rises, falls, and number of runs in random sequences

TL;DR

This paper develops a matrix-based recursive method to derive explicit generating functions for the joint distribution of rises, falls, and runs in multivariate random sequences, advancing the understanding of their probabilistic structure.

Contribution

It introduces a novel matrix formulation and recursive approach to explicitly compute joint distributions of runs, rises, and falls in multivariate sequences.

Findings

Derived explicit generating functions for joint distributions

Established recursive relations for multivariate sequences

Provided formulas for arbitrary specifications of rises and falls

Abstract

By using the matrix formulation of the two-step approach to the distributions of runs, a recursive relation and an explicit expression are derived for the generating function of the joint distribution of rises and falls for multivariate random sequences in terms of generating functions of individual letters, from which the generating functions of the joint distribution of rises, falls, and number of runs are obtained. An explicit formula for the joint distribution of rises and falls with arbitrary specification is also obtained.