Asymptotic Period of an Aperiodic Markov Chain

#### E.A. van Doorn

2018, v.24, Issue 5, 759-778

ABSTRACT

We introduce the concept of {\em asymptotic period\/} for an irreducible and
aperiodic, discrete-time Markov chain $\mathcal{X}$ on a countable state
space, and develop the theory leading to its formal definition. The asymptotic
period of $\mathcal{X}$ equals one -- its period -- if $\mathcal{X}$ is
recurrent, but may be larger than one if $\mathcal{X}$ is transient;
$\mathcal{X}$ is {\em asymptotically aperiodic\/} if its asymptotic period equals
one. Some sufficient conditions for asymptotic aperiodicity are presented. The
asymptotic period of a birth-death process on the nonnegative integers is
studied in detail and shown to be equal to 1, 2 or $\infty$. Criteria for the
occurrence of each value in terms of the 1-step transition probabilities are
established.

Keywords: aperiodicity, birth-death process, harmonic function, period, transient Markov chain, transition probability