# On strong stationary times and approximation of Markov chain hitting times by geometric sums

## TL;DR

This paper extends the understanding of Markov chain hitting times by providing bounds on their approximation via geometric sums of IID variables, using strong stationary times for improved accuracy.

## Contribution

It introduces explicit bounds for approximating hitting times with geometric sums, utilizing strong stationary times, and discusses their construction and properties.

## Key findings

- Derived bounds on the distance between hitting times and geometric sum approximations.
- Showed how strong stationary times can be used to improve hitting time approximations.
- Provided methods for constructing the approximating distributions.

## Abstract

Consider a discrete time, ergodic Markov chain with finite state space which is started from stationarity. Fill and Lyzinski (2014) showed that, in some cases, the hitting time for a given state may be represented as a sum of a geometric number of IID random variables. We extend this result by giving explicit bounds on the distance between any such hitting time and an appropriately chosen geometric sum, along with other related approximations. The compounding random variable in our approximating geometric sum is a strong stationary time for the underlying Markov chain; we also discuss the approximation and construction of this distribution.