Distribution of a Markov chain in reverse-time with cluster observations in the extremes of a finite time window

TL;DR

This paper derives formulas for the distribution and transition probabilities of a Markov chain observed in reverse time within a finite window, focusing on cluster observations at the extremes.

Contribution

It introduces explicit formulas for reverse-time distributions and transition matrices of Markov chains with extreme cluster observations within a finite time window.

Findings

Formulas for reverse-time distribution of Markov chains

Transition probability matrices in reverse-time setting

Application to cluster observations at time window extremes

Abstract

In this brief note, we find formulas for the distribution and the transition probability matrices of a stochastic process described as a time-reversion in a finite time window of a Markov chain, with cluster observation of the Markov state in the extremes of that window.