Waiting times and stopping probabilities for patterns in Markov chains

TL;DR

This paper develops a straightforward method to compute the average waiting time and the probabilities of stopping at specific patterns in finite Markov chains, facilitating analysis of pattern occurrences.

Contribution

It introduces an easy approach for calculating mean waiting times and stopping probabilities for pattern occurrences in Markov chains.

Findings

Provides formulas for mean waiting times for pattern occurrences.

Derives probabilities of stopping at specific patterns.

Simplifies analysis of pattern detection in Markov processes.

Abstract

Suppose that $\mathcal C$ is a finite collection of patterns. Observe a Markov chain until one of the patterns in $\mathcal C$ occurs as a run. This time is denoted by $\tau$. In this paper, we aim to give an easy way to calculate the mean waiting time $E(\tau)$ and the stopping probabilities $P(\tau=\tau_A)$ with $A\in\mathcal C$, where $\tau_A$ is the waiting time until the pattern $A$ appears as a run.