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

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.

Improved Method for Approximating the Hazard Rate for Convolution Poisson Random Variables

“…The hazard rate function for a discrete distribution taking values in ℤ + = {0,1,2,3, … } is defined as (see, e.g., Daly [10]). Consider the same example of the sum of two independent Poisson random variables with means 1 and 2 as in the previous section.…”