Distribution of a Unified $(k_1,k_2,\ldots,k_m)$-run

TL;DR

This paper investigates the distribution and waiting times of a generalized run pattern in multi-state trials, unifying various existing patterns and demonstrating applications in DNA pattern analysis.

Contribution

It introduces a unified framework for $(k_1,k_2, ightarrow,k_m)$-runs, generalizing multiple known run patterns and deriving their distributional properties.

Findings

Unified distributional formulas for generalized runs.

Special cases recover known run distributions.

Application to DNA pattern frequency analysis.

Abstract

We explore a unified $(k_1,k_2,\ldots,k_m)$-run in multi-state trials, examining its distributional properties and waiting time distribution. Our study reveals that this particular run serves as a generalization encompassing various patterns. Additionally, we discuss various results pertaining to existing patterns as special cases. To illustrate our findings, we provide an application related to DNA frequent patterns.