Upward and Downward Runs on Partially Ordered Sets

TL;DR

This paper studies Markov chains on partially ordered sets, providing conditions for recurrence and transience, and deriving invariant distributions, with applications to various special cases like trees and semigroups.

Contribution

It introduces new conditions for recurrence and transience of Markov chains on posets and derives explicit invariant distributions for these chains.

Findings

Conditions for recurrence and transience are established.

Explicit formulas for invariant distributions are provided.

Applications to special poset structures like trees and semigroups.

Abstract

We consider Markov chains on partially ordered sets that generalize the success-runs and remaining life chains in reliability theory. We find conditions for recurrence and transience and give simple expressions for the invariant distributions. We study a number of special cases, including rooted trees, uniform posets, and posets associated with positive semigroups.