An orthogonal-polynomial approach to first-hitting times of birth-death processes

In a recent paper in the Journal of Theoretical Probability Gong, Mao and Zhang, using the theory of Dirichlet forms, extended Karlin and McGregor's classical results on first-hitting times of a birth-death process on the nonnegative integers by establishing a representation for the Laplace transform $\mathbb{E}[e^{sT_{ij}}]$ of the first-hitting time $T_{ij}$ for $any$ pair of states $i$ and $j$, as well as asymptotics for $\mathbb{E}[e^{sT_{ij}}]$ when either $i$ or $j$ tends to infinity. It will be shown here that these results may also be obtained by employing tools from the orthogonal-polynomial toolbox used by Karlin and McGregor, in particular $associated$ $polynomials$ and $Markov's$ $theorem$.

satisfy both the backward and forward Kolmogorov equations, and mostly also that they are uniquely determined by the birth rates λ i and death rates µ i . Karlin and McGregor [14] have shown that the latter is equivalent to assuming ∞ n=0 π n + 1 λ n π n = ∞, where the π n are constants given by π 0 := 1 and π n := λ 0 λ 1 . . . λ n−1 µ 1 µ 2 . . . µ n , n > 0.
We note that condition (1) does not exclude the possibility of explosion, escape from S, via all states larger than the initial state, to an absorbing state ∞.
The representation (3) was observed explicitly for the first time by Karlin and McGregor themselves in [16,Page 378]. Since then several authors have rediscovered the result or provided alternative proofs (see Diaconis and Miclo [4] for some references). In a recent paper in this journal Gong, Mao and Zhang [12], using the theory of Dirichlet forms, extended Karlin and McGregor's result by establishing a representation for the Laplace transform of the first-hitting time T ij for any pair of states i = j, as well as asymptotics when either i or j tends to infinity. It will be shown here that these results may also be obtained by first-hitting times of birth-death processes 3 exploiting Karlin and McGregor's toolbox, which is the theory of orthogonal polynomials.
Our findings, which are actually somewhat more general than those of Gong, Mao and Zhang, are presented in Section 3 and proven in Section 4. In the next section we introduce some further notation, terminology and preliminary results. Since a path between two states in a birth-death process has to hit all intermediate states, we obviously havê So for notational simplicity -and without loss of generality -we will restrict ourselves to an analysis of T 0n and T n0 for n > 0.

Preliminaries
We will use the shorthand notation and, following Anderson [1,Chapter 8], We have K ∞ + L ∞ = ∞ by our assumption (1), while, obviously, Also, C + D = K ∞ L ∞ , so (1) is actually equivalent to C + D = ∞. Whether the quantities C and D are infinite or not determines the type of the boundary at infinity (see, for example, Anderson [1, Section 8.1]), but also, as we shall see, the asymptotic behaviour of the polynomials Q n of (4). Since the birth-death polynomials Q n satisfy the three-terms recurrence relation (4), they are orthogonal with respect to a positive Borel measure on the nonnegative real axis, and have positive and simple zeros. The orthogonalizing measure for the polynomials Q n (normalized to be a probability measure) is not necessarily uniquely determined by the birth and death rates, but there exists, in any case, a unique natural measure ψ, characterized by the fact that the minimum of its support is maximal. We refer to Chihara's book [3] for properties of orthogonal polynomials in general, and to Karlin and McGregor's papers [14] and [15] for results on birth-death polynomials in particular (see also [10, Section 3.1] for a concise overview). For our purposes the following properties of birth-death polynomials are furthermore relevant.
With x n1 < x n2 < . . . < x nn denoting the n zeros of Q n (x), there is the classical separation result 0 < x n+1,i < x ni < x n+1,i+1 , i = 1, 2, . . . , n, n ≥ 1, so that the limits exist. We further let σ := lim (possibly infinity). The numbers ξ i may be defined alternatively as where supp stands for support. So knowledge of the (natural) orthogonalizing measure for the polynomials Q n implies knowledge of the numbers ξ i . It is clear from the definition of ξ i that Moreover we have, for all i ≥ 1, as is evident from the alternative definition of ξ i . By suitably interpreting [6, Equations (2.6) and (2.11)] it follows that where the left-hand side should be interpreted as infinity if ξ 1 = 0. In particular, Also, by [6,Theorem 2], and Given a sequence of birth-death polynomials {Q n } we obtain the sequence {Q (l) n } of associated polynomials of order l ≥ 0 by replacing Q n by Q (l) n , λ n by λ n+l and µ n by µ n+l in the recurrence relation (4). Evidently, the polynomials first-hitting times of birth-death processes 5 Q (l) n are birth-death polynomials again, so Q (l) n (x) has simple, positive zeros x nn and we can write while it follows by induction that where L −1 := 0. Note that Q i and σ (l) in analogy to (7) and (8) so that σ (l) = σ, l ≥ 0.
Moreover, [5, Theorem 1] tells us that Since the polynomials Q (l) n are birth-death polynomials they are orthogonal with respect to a unique natural (probability) measure ψ (l) on the nonnegative real axis. A key ingredient in our analysis is Markov's Theorem, which relates the Stieltjes transform of the measure ψ (l) to the polynomials Q (l) We note that ψ (l) is not necessarily the only orthogonalizing measure for the polynomials Q (l) n , a setting usually not covered in statements of Markov's Theorem in the literature (see, for example, [3, Page 89]). However, an extension of the original theorem that serves our needs can be found in Berg [2] (see in particular [2, Section 3], where the measure µ (0) corresponds to our ψ (l) ).
We will also have use for a classical result in the theory of continued fractions relating the Stieltjes transforms of the measures ψ (l) and ψ (l+1) , namely, Again we refer to Berg [2, Section 4] for statements of this result in the generality required in our setting.
Our final preliminary results concern asymptotics for the polynomials Q (l) n as n → ∞, which may be obtained by suitably interpreting the results of [17] (which extend those of [6]). We state the results in three propositions and give more details about their derivations in Section 4. Recall that ξ an entire function with simple, positive zeros ξ an entire function with simple zeros ξ 1 = 0 and ξ i+1 > 0, i ≥ 1; (iv) if D < ∞, for l = 0 and µ 0 > 0, or l ≥ 1, an entire function with simple, positive zeros ξ n are collected in the first theorem. Theorem 1 We have, for µ 0 ≥ 0 and n ≥ 1, and, if C + D = ∞, 1 . By choosing s = 0 in (16) and (17) and using (11), we obtain expressions for the probabilities P(T 0n < ∞) and P(T n0 < ∞) that are in accordance with [15, an unnumbered formula on page 387 and Theorem 10]. For convenience we state the results as a corollary of Theorem 1, but remark that a proof of (19) on the basis of (17) would require additional motivation in the case ξ and, if C + D = ∞, After a little algebra (17) and (11)  Subsequently applying Propositions 1, 2 (iii) and 3 (iv) we obtain the second corollary of Theorem 1.
Corollary 2 If C + D = ∞, but C < ∞ or D < ∞, then, for n ≥ 1, where the infinite products are entire functions with simple, positive zeros ξ (n+1) i and ξ i , i ≥ 1. and The infinite products in (21) and (22) are reciprocals of entire functions with simple, positive zeros ξ i and ξ (1) i , i ≥ 1, respectively.
first-hitting times of birth-death processes 9 4 Proofs

Proofs of Propositions 1-3
The conclusions regarding C and D in the Propositions 1, 2 and 3 are given already in (6), while the statements (i) in Propositions 2 and 3 are implied by (11). The other statements follow from results in [17], where two casescorresponding in the setting at hand to µ 0 = 0 and µ 0 > 0 -are considered simultaneously by means of a duality relation involving polynomials R n and R * n . The asymptotic results for R n may be translated into asymptotics for Q n if µ 0 = 0, while the results for R * n , suitably interpreted, give asymptotics for Q n if µ 0 > 0, and for Q
To obtain (17) we note that [15, Equation (3.21)]) also yieldŝ which upon substitution in (2) leads tô Moreover, by Karlin and McGregor's representation formula for the transition probabilities P ij (t) (see [14, Section III.6]) we have P 00 (t) = ∞ 0 e −xt ψ(dx), where ψ is a (probability) measure with respect to which the polynomials Q n are orthogonal. Since the condition C + D = ∞ is equivalent to (1), it ensures that the transition probabilities are uniquely determined by the birth and death rates, whence ψ must be the natural measure (see [14]). So we havê Subsequently applying (15) with l = 0, it follows that whence, more generally, and, by analytic continuation, Since T n0 = T n,n−1 + · · · + T 10 , while T n,n−1 , . . . , T 10 are independent random variables, Markov's Theorem (14) implies that we can write .
Recalling (12) we conclude that this expression holds for s < ξ 1 .

Proof of Theorem 2
Letting n → ∞ in (16) and applying the results of Propositions 1, 2 and 3 readily yields the first statement of Theorem 2.

Concluding remarks
First we note that the result (24) -or rather a generalization of (24) -may be derived directly from the Kolmogorov differential equations and Karlin and McGregor's representation formula for the transition probabilities P ij (t). The argument is given on [8, Page 508] (and essentially already on [15,Page 385]) and yields Note that as a consequence of (17) and (26) we have, for all m ≥ 0, which implies a partial extension of Markov's Theorem (14) to the effect that, for m ≥ 0 and l ≥ 1 (and l = 0 if µ 0 > 0), Using (14), (15), and the recurrence relation for the polynomials Q which is consistent with [15, Equations (9.9) and (9.14)] (also when ξ 1 = 0). If we do not impose the condition C + D = ∞, the birth and death rates do not necessarily determine a birth-death process uniquely. However, as observed in [12], several results remain valid if C +D < ∞, provided they are interpreted as properties of the minimal process, which is the process with an absorbing boundary at infinity (and which is always associated with the natural measure for the polynomials Q n , see [7]). Concretely, if C + D < ∞ the arguments leading to Theorem 1, and hence Theorem 1 itself and Corollary 1, remain valid. Moreover, the results in [17] imply that, for l ≥ 0,  (16) and (20) we readily conclude that the results in Theorem 2 for C < ∞, D = ∞ are actually valid for C + D < ∞ as well.
In the setting C + D < ∞ Gong, Mao and Zhang [12] pay attention also to the maximal process, the process that is characterized by a reflecting barrier at infinity. In this case the measure featuring in the representation for P 00 (t), and hence in (23), is not the natural measure. Although, applying the results of [7], the relevant measure can be identified and expressed in terms of a natural measure corresponding to a dual birth-death process, application of Markov's Theorem does not seem feasible in this case.
(Since, by [9, Theorem 3.1], the condition above is equivalent to ξ 1 -recurrence of the process, this result may also be obtained by applying [13, Lemma 3.3.3 (iii)] to the setting at hand, see [11,Lemma 3.2].) It now follows from (22) that If µ 0 = 0 then ξ 1 = 0, so the result does not take us by surprise, but for µ 0 > 0 we regain an interesting extension of Proposition 3 (iv) -recently obtained by Gao and Mao [11, Lemma 3.4] -since it has consequences for the existence of quasi-stationary distributions.