Hitting Times and Interlacing Eigenvalues: A Stochastic Approach Using Intertwinings

Abstract

Abstract. We develop a systematic matrix-analytic approach, based on intertwinings of Markov semigroups, for proving theorems about hitting-time distributions for finite-state Markov chains-an approach that (sometimes) deepens understanding of the theorems by providing corresponding sample-path-by-sample-path stochastic constructions. We employ our approach to give new proofs and constructions for two theorems due to Mark Brown, theorems giving two quite different representations of hitting-time distributions for finite-state Markov chains started in stationarity. The proof, and corresponding construction, for one of the two theorems elucidates an intriguing connection between hitting-time distributions and the interlacing eigenvalues theorem for bordered symmetric matrices.

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

“…Using Lemma 1.2, Theorem 1.3 below gives an explicit link between strong stationary times and geometric sums. Its conclusion is the same as that of Theorem 4.2 of Fill and Lyzinski [6], though the stated conditions are somewhat stronger (see Remark 1.4 below), and the proof uses different techniques. We give it here to motivate the main results of this paper that will follow in Section 2.…”

“…Our primary interest in this note is in the time it takes X, when initialised in the stationary regime, to hit a given state j ∈ S. Many structural results are known about such hitting times. In particular, it is known that, under certain conditions, this hitting time may be expressed as a geometric sum of independent and identically distributed (IID) random variables; see Theorem 4.2 of Fill and Lyzinski [6] for a precise statement. Our main goal in this note is to consider the approximation of such a hitting time by a geometric sum.…”

“…Our main goal in this note is to consider the approximation of such a hitting time by a geometric sum. Motivated by the results of [6], an appropriate choice for the compounding distribution in the approximating geometric sum is that of a strong stationary time for the underlying Markov chain X with an appropriate initial distribution. Recall that a strong stationary time, T , for X is a randomized stopping time such that X T ∼ π, and X T is independent of T .…”

“…Remark 1.4. Theorem 4.2 of Fill and Lyzinski [6] also gives conditions under which the hitting time W j is distributed as a geometric sum. Their result is stronger, as they work under the assumption that P t (j, j) is decreasing in t in the place of our assumption that π y P(X t = j) ≤ π j P(X t = y) for all t and y. Lemma 1.2 shows that our assumption is more restrictive than that of Fill and Lyzinski.…”

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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.

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