Multiple consecutive runs of multi-state trials: distributions of $(k_1, k_2, \dots, k_\ell)$ patterns

TL;DR

This paper derives the distribution of complex multi-state patterns in random sequences, extending previous methods to handle arbitrary pattern lengths and providing numerical examples for illustration.

Contribution

It introduces a recursive method to compute distributions of multi-state patterns of arbitrary length in random sequences, advancing prior combinatorial approaches.

Findings

Distribution formulas for arbitrary pattern lengths

Recursive computational method for pattern distributions

Numerical examples demonstrating the results

Abstract

The pattern $(k_1, k_2, \dots, k_\ell)$ is defined to have at least $k_1$ consecutive $1$'s followed by at least $k_2$ consecutive $2$'s, $\dots$, followed by at least $k_\ell$ consecutive $\ell$'s. By iteratively applying the method that was developed previously to decouple the combinatorial complexity involved in studying complicated patterns in random sequences, the distribution of pattern $(k_1, k_2, \dots, k_\ell)$ is derived for arbitrary $\ell$. Numerical examples are provided to illustrate the results.