Stein's Method for Compound Geometric Approximation

Abstract

We apply Stein's method for probabilistic approximation by a compound geometric distribution, with applications to Markov chain hitting times and sequence patterns. Bounds on our Stein operator are found using a complex analytical approach based on generating functions and Cauchy's formula.

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

“…We use the remainder of this section to introduce some necessary background material on strong stationary times, and to provide an explicit link between geometric sums of strong stationary times and Markov chain hitting times. Section 2 then contains our main approximation results, and a comparison of the present results with those of Daly [3], who has previously considered the approximation of Markov chain hitting times by geometric sums. Note, however, that the approximating geometric sum chosen in [3] is very different to that used here, and that the results of [3] have some deficiencies which the present results remedy (for example, the bounds in [3] offer no guarantee of sharpness).…”

“…Section 2 then contains our main approximation results, and a comparison of the present results with those of Daly [3], who has previously considered the approximation of Markov chain hitting times by geometric sums. Note, however, that the approximating geometric sum chosen in [3] is very different to that used here, and that the results of [3] have some deficiencies which the present results remedy (for example, the bounds in [3] offer no guarantee of sharpness). We also note that the approximating geometric sum we choose here stochastically dominates our hitting time, and explore implications of this in Section 2.…”

“…Proof. Using the characterization of geometric sums presented in Section 2 of Daly [3], and the fact that P(W j = 0) = π j , the conclusion follows on showing that…”

“…where the infimum is taken over all couplings of W j and U. Following the work of Daly [3], to derive such a bound we construct an integer-valued random variable Y (which may depend on T (j) ) such that Y + T (j) ≥ 0 and…”

“…Remark 2.1. The results of [3] are stated under the assumption that the support of T (j) is bounded. This assumption may be removed using by making suitable adjustments to the proof of Theorem 2.1 of that paper.…”