On the Strong Ratio Limit Property for Discrete-Time Birth-Death Processes

A sufficient condition is obtained for a discrete-time birth-death process to possess the strong ratio limit property, directly in terms of the one-step transition probabilities of the process. The condition encompasses all previously known sufficient conditions.


Introduction
In what follows X ≡ {X(n), n = 0, 1, . . .} is a discrete-time birth-death process on N ≡ {0, 1, . . . }, with tridiagonal matrix of one-step transition probabilities Following Karlin and McGregor [6] we will refer to X as a random walk. We assume throughout that p j > 0, q j+1 > 0, r j ≥ 0, and p j + q j + r j = 1 for j ≥ 0, where q 0 := 0. We let π 0 := 1, π n := p 0 p 1 · · · p n−1 q 1 q 2 · · · q n , n ≥ 1, (1.1) and define the polynomials Q n via the recurrence relation xQ n (x) = q n Q n−1 (x) + r n Q n (x) + p n Q n+1 (x), n > 1, Karlin and McGregor [6] have shown that the n-step transition probabilities P ij (n) := Pr{X(n) = j | X(0) = i} = (P n ) ij , n ≥ 0, i, j ∈ N , may be represented in the form P ij (n) = π j [−1 ,1] x n Q i (x)Q j (x)ψ(dx), (1.3) This paper is a contribution to the Special Issue on Orthogonal Polynomials, Special Functions and Applications (OPSFA14). The full collection is available at https://www.emis.de/journals/SIGMA/OPSFA2017.html where ψ is the (unique) Borel measure on the interval [−1, 1], of total mass 1 and with infinite support, with respect to which the polynomials Q n are orthogonal. Adopting the terminology of [3] we will refer to the measure ψ as a random walk measure. Of particular interest to us is η := sup supp(ψ), the largest point in the support of the random walk measure ψ, which may also be characterized in terms of the polynomials Q n by for all n ≥ 0 (1.4) (see, for example, Chihara [1,Theorem II.4.1]). We will see in the next section that η > 0. The random walk X is said to have the strong ratio limit property (SRLP) if the limits exist simultaneously. The SRLP was introduced in the more general setting of discrete-time Markov chains on a countable state space by Orey [8] and Pruitt [9], but the problem of finding conditions for the limits (1.5) to exist in the specific setting of random walks had been considered before in [6]. A satisfactory and comprehensive solution to the problem of finding conditions for the SRLP is still lacking, even in the relatively simple setting at hand. So it remains a challenge to find necessary and/or sufficient conditions. For more information on the history of the problem we refer to [5] and [7]. In [5, Theorem 3.1] a necessary and sufficient condition for the random walk X to have the SRLP has been given in terms of the associated random walk measure ψ. Namely, letting in which case we have Note that the denominator in (1.6) is positive since η > 0, so that C n (ψ) exists and is nonnegative for all n. Some sufficient conditions for (1.7) -and, hence, for X to possess the SRLP -are also given in [5]. In particular, [5, Theorem 3.2] tells us that The reverse implication is conjectured in [5] to be valid as well.
In this paper we will prove a sufficient condition for X to have the SRLP directly in terms of the one-step transition probabilities. Concretely, we will establish the following result.
Together with (1.8) this result immediately leads to the following.
Theorem 1.2. If the random walk X satisfies (1.9) then X possesses the SRLP.
We will see that Theorem 1.2 encompasses all previously obtained sufficient conditions for the SRLP.
The proof of Proposition 1.1 will be based on three lemmas. Lemma 2.1 and a number of preliminary results related to the polynomials Q n and the orthogonalizing measure ψ are collected in the next section. Two further auxiliary lemmas are established in Section 3. The actual proof of Proposition 1.1 and some concluding remarks can be found in Section 4, which also contains an example showing that (1.9) is not necessary for the SRLP.

Preliminaries
Whitehurst [11,Theorem 1.6] has shown that the random walk measure ψ satisfies The measure ψ is symmetric about 0 if (and only if) the random walk X is periodic, that is, if r j = 0 for all j (see [6, p. 69]). In this case we also have n ≥ 0, and it follows from (1.3) that P ij (n) = 0 if n + i + j is odd. Hence the limits in (1.5) will not exist if X is periodic, which is also reflected by the fact that C n (ψ) = 1 for all n in this case.
The above lemma also plays a central role in [2], where the conditions in (2.5) are shown to be equivalent to asymptotic aperiodicity of the random walk. For completeness' sake we have included the proof.
We recall from [6] that X is recurrent, that is, the probability, for any state, of returning to that state is one, if and only if X is called transient if it is not recurrent. It has been shown in [6] that so we must have η = 1 if X is recurrent. From Lemma 2.1 we now obtain X is aperiodic and recurrent ⇒ lim

Two auxiliary lemmas
Throughout this section θ is a fixed number satisfying θ ≥ η. Defining q 0 (θ) := 0 and the parameters p j (θ), q j (θ) and r j (θ) satisfy p j (θ) > 0, q j+1 (θ) > 0, r j (θ) ≥ 0, and p j (θ) + q j (θ) + r j (θ) = 1, so that they may be interpreted as the one-step transition probabilities of a random walk X θ on N . Denoting the corresponding polynomials by Q n (·; θ) it follows readily that so that the associated measure ψ θ satisfies Evidently, we have while the analogues π n (θ) of the constants π n of (1.1) are easily seen to satisfy π n (θ) = π n Q 2 n (θ), n ≥ 0. (3.3) (In [4, Appendix 2]) the special case θ = η is considered.) Obviously, X θ is periodic if and only if X is periodic. Note that by choosing θ = 1 we return to the setting of the previous sections. We have seen in Lemma 2.1 that (−1) n Q n (−1; θ) is increasing, and strictly increasing for n sufficiently large, if X θ is aperiodic, or, equivalently, X is aperiodic. It thus follows from (3.2) that |Q n (θ)/Q n (−θ)| is decreasing, and strictly decreasing for n sufficiently large, if X is aperiodic. Since Q n (−x; θ) = (−1) n Q n (x; θ) if X θ is periodic, we conclude the following.
Lemma 3.1. Let θ ≥ η. If X is periodic then |Q n (θ)/Q n (−θ)| = 1 for all n. If X is aperiodic then |Q n (θ)/Q n (−θ)| is decreasing and tends to a limit satisfying In what follows we let so that in particular M ∞ (1) equals the left-hand side of (1.9). In combination with Lemma 2.1, interpreted in terms of X θ , Lemma 3.1 gives us the next result.
In view of (1.8) it follows in particular that the random walk X possesses the SRLP if M ∞ (η) = ∞, which readily leads to some further sufficient conditions. Indeed, choosing θ = η and defining L(η) in analogy with (2.7) we have so, in analogy with (2.8), Corollary 3.2 leads to X is aperiodic and L(η) = ∞ ⇒ lim n→∞ |Q n (η)/Q n (−η)| = 0. (3.6) By (2.3) we have L(η) ≥ L(1) ≡ L so the premise in (3.6) certainly prevails if X is aperiodic and recurrent. When L(η) = ∞ the random walk X is called η-recurrent (see [4] for more information). The conclusion that η-recurrence is sufficient for an aperiodic random walk to possess the SRLP is not surprising, since Pruitt [9, Theorem 2] already established this result in the more general setting of symmetrizable Markov chains. Another sufficient condition for the conclusion in (3.5) is obtained in analogy with (2.10), namely Since, by (2.3), Q j+1 (η) ≤ Q j (η) it follows in particular that which, by Corollary 3.2, leads to Proposition 1.1.
It seems unlikely that there are values of θ 1 and θ 2 such that η < θ 1 < θ 2 and M ∞ (θ 1 ) = ∞, but M ∞ (θ 2 ) < ∞, since there do not seem to be values of x > η that are "special" in any sense. So we conjecture that M ∞ (θ 1 ) and M ∞ (θ 2 ) converge or diverge together. It is tempting to go one step further by extending this conjecture to η ≤ θ 1 < θ 2 . Maintaining the conjecture in [5] that also the reverse implication in (1.8) is valid, we would then arrive at the conjecture that (1.9) is not only sufficient but also necessary for X to possess the SRLP. However, this not correct, since it is possible to have M ∞ (η) = ∞ and M (1) < ∞ simultaneously, as the next example shows.
Example 4.1. Consider a random walkX determined by one-step transition probabilitiesp j ,q j andr j withr 0 > 0 andr j = 0 for j > 0. Quantities associated withX will be indicated by a tilde. We will assume thatX is recurrent, so thatη = 1. Now let α > 1 and define These quantities, like those in (3.1), can be interpreted as the one-step transition probabilities of a new random walk X , say. In what follows we associate quantities without tilde with X . In analogy with (3.2) and (3.3) we thus have Q n (x) =Q n (αx)/Q(α) and π n =π nQ 2 n (α). Also, η =ηα −1 = α −1 < 1, so that X must be transient. Next, letting M n (θ) be defined as in (3.4) and (3.1) where p j , q j and r j are given by (4.1), we have M ∞ (1) = r 0 j≥0 1 p j π j < ∞, since X is transient. But on the other hand sinceX is recurrent.
We have already encountered several known sufficient conditions for the random walk X to possess the SRLP. In particular, η-recurrence -and thus recurrence, which is simply 1-recurrence -was shown to be sufficient in (3.6). Also, in view of (2.9) we regain directly from Theorem 1 Karlin and McGregor's claim on [6, p. 77] j≥0 r j p j = ∞ ⇒ X possesses the SRLP, referred to after (3.7). Several authors (see [6, p. 77], [5,Corollary 3.2]) have shown that for the SRLP to prevail it is sufficient that r j > δ > 0 for j sufficiently large, but this condition is evidently weaker than the previous one.