# An Orthogonal-Polynomial Approach to First-Hitting Times of Birth–Death Processes

## Abstract

In a recent paper in this journal, 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*.

### Keywords

Birth–death process First-hitting time Orthogonal polynomials Associated polynomials Markov’s theorem### Mathematics Subject Classification (2010)

60J80 42C05## 1 Introduction

*birth–death process*is a continuous-time Markov chain \(\mathcal {X} := \{X(t),~t \ge 0\}\) taking values in \(S := \{0,1,2,\ldots \}\) with

*q*-matrix \(Q := (q_{ij},~i,j \in S)\) given by

*S*, via state 0, to an absorbing state \(-1\). Throughout this paper, we will assume that the transition probabilities

*birth rates*\(\lambda _i\) and

*death rates*\(\mu _i\). Karlin and McGregor [14] have shown that the latter is equivalent to assuming

*explosion*, escape from

*S*, via all states larger than the initial state, to an absorbing state \(\infty \).

*j*, starting in state \(i\ne j\). Then, writing

*birth–death polynomials*associated with the process \(\mathcal {X}\), that is, the \(Q_n\) satisfy the recurrence relation

In a recent paper in this journal, Gong et al. [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 \(i\ne j\), 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 exploiting Karlin and McGregor’s toolbox, which is the theory of orthogonal polynomials.

## 2 Preliminaries

*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 behavior 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 \(\psi \), 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, 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.

*n*zeros of \(Q_n(x),\) there is the classical separation result

*support*. So knowledge of the (natural) orthogonalizing measure for the polynomials \(Q_n\) implies knowledge of the numbers \(\xi _i\). It is clear from the definition of \(\xi _i\) that

*associated polynomials*of order \(l\ge 0\) by replacing \(Q_n\) by \(Q_n^{(l)}\), \(\lambda _n\) by \(\lambda _{n+l}\) and \(\mu _n\) by \(\mu _{n+l}\) in the recurrence relation (4). Evidently, the polynomials \(Q_n^{(l)}\) are birth–death polynomials again, so \(Q_n^{(l)}(x)\) has simple, positive zeros \(x_{n1}^{(l)}<x_{n2}^{(l)}<\dots <x_{nn}^{(l)}\) and we can write

*n*if \(\mu _0=0\).

*natural*(probability) measure \(\psi ^{(l)}\) on the nonnegative real axis. A key ingredient in our analysis is

*Markov’s theorem*, which relates the

*Stieltjes transform*of the measure \(\psi ^{(l)}\) to the polynomials \(Q_n^{(l)}\) and \(Q_n^{(l+1)}\), namely

Our final preliminary results concern asymptotics for the polynomials \(Q_n^{(l)}\) as \(n\rightarrow \infty \), 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 Sect. 4. Recall that \(\xi ^{(l)}_0=-\infty \) and \(Q_n(0)=1\) if \(\mu _0=0\).

**Proposition 1**

**Proposition 2**

- (i)for \(l\ge 0\),$$\begin{aligned} \lim _{n\rightarrow \infty } Q_n^{(l)}(0) = 1 + \mu _l\pi _l(L_\infty - L_{l-1}) < \infty ; \end{aligned}$$
- (ii)if \(C=\infty \), for \(l\ge 0\),$$\begin{aligned} \lim _{n\rightarrow \infty } Q_n^{(l)}(x) = \left\{ \begin{array}{ll} \infty &{}\quad \text {if}~~x < 0\\ 0 &{}\quad \text {if}~~0 < x \le \xi _k^{(l)} \text {~for~some~}k\ge 1; \end{array} \right. \end{aligned}$$
- (iii)if \(C<\infty \), for \(l \ge 0\),an entire function with simple, positive zeros \(\xi _i^{(l)},~i\ge 1\).$$\begin{aligned} \lim _{n\rightarrow \infty } \frac{Q_n^{(l)}(x)}{Q_n^{(l)}(0)} = \prod _{i=1}^\infty \left( 1-\frac{x}{\xi _i^{(l)}}\right) , \quad x \in \mathbb {R}, \end{aligned}$$

**Proposition 3**

- (i)for \(l=0\) and \(\mu _0>0\), or \(l \ge 1\),$$\begin{aligned} \lim _{n\rightarrow \infty } Q_n^{(l)}{(0)} = \infty ; \end{aligned}$$
- (ii)if \(D=\infty \), for \(l \ge 0\),$$\begin{aligned} \lim _{n\rightarrow \infty } Q_n^{(l)}(x) = \left\{ \begin{array}{ll} \infty &{}\quad \text {if}~~\xi ^{(l)}_{2k} < x \le \xi ^{(l)}_{2k+1} \text {~for~some~}k\ge 0\\ -\infty &{}\quad \text {if}~~\xi ^{(l)}_{2k+1} < x \le \xi ^{(l)}_{2k+2} \text {~for~some~}k\ge 0; \end{array} \right. \end{aligned}$$
- (iii)if \(D<\infty \) and \(\mu _0=0\),an entire function with simple zeros \(\xi _1=0\) and \(\xi _{i+1}>0,~i\ge 1\);$$\begin{aligned} \lim _{n\rightarrow \infty } \frac{Q_n(x)}{L_{n-1}} = -x K_\infty \prod _{i=1}^\infty \left( 1-\frac{x}{\xi _{i+1}}\right) , \quad x \in \mathbb {R}, \end{aligned}$$
- (iv)if \(D<\infty \), for \(l=0\) and \(\mu _0>0\), or \(l \ge 1\),an entire function with simple, positive zeros \(\xi _i^{(l)},~i\ge 1\).

## 3 Results

Representations for \({\mathbb {E}}[e^{sT_{0n}}{\mathbb {I}}_{\{T_{0n}<\infty \}}]\) and \({\mathbb {E}}[e^{sT_{n0}}{\mathbb {I}}_{\{T_{n0}<\infty \}}]\) in terms of the polynomials \(Q_n^{(l)}\) are collected in the first theorem.

**Theorem 1**

Note that for \(s<0\), we have \({\mathbb {E}}[e^{sT_{0n}}{\mathbb {I}}_{\{T_{0n}<\infty \}}] ={\mathbb {E}}[e^{sT_{0n}}]\), so the representation (16) reduces to Karlin and McGregor’s result (3). The explicit representation (17) is new, but may be obtained by a limiting procedure from Gong et al. [12, Corollary 3.6], where a finite state space is assumed.

By choosing \(s=0\) in (16) and (17) and using (11), we obtain expressions for the probabilities \(\mathbb {P}(T_{0n}<\infty )\) and \(\mathbb {P}(T_{n0}<\infty )\) 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 \(\xi _1^{(1)} = 0\).

**Corollary 1**

**Corollary 2**

Assuming a denumerable state space, but under the condition \(C=\infty \) and \(D<\infty \), Gong et al. give in [12, Theorem 5.5 (a)] a representation for \({\mathbb {E}}[e^{sT_{n0}}],~s<0\), which is encompassed by Corollary 2. Indeed, in this case we have \(L_\infty = \infty \), and hence, by (19), \(\mathbb {P}(T_{n0}<\infty ) = 1\).

Asymptotic results for \({\mathbb {E}}[e^{sT_{0n}}{\mathbb {I}}_{\{T_{0n}<\infty \}}]\) and \({\mathbb {E}}[e^{sT_{n0}}{\mathbb {I}}_{\{T_{n0}<\infty \}}]\) as \(n\rightarrow \infty \) are summarized in the second theorem.

**Theorem 2**

## 4 Proofs

### 4.1 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 cases—corresponding in the setting at hand to \(\mu _0=0\) and \(\mu _0>0\)—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 \(\mu _0=0\), while the results for \(R_n^*\), suitably interpreted, give asymptotics for \(Q_n\) if \(\mu _0>0\), and for \(Q_n^{(l)}\) with \(l\ge 1\). Concretely, the statements in Proposition 1, Proposition 2 (ii) and Proposition 3 (ii) regarding the case \(x<0\) follow from [17, Lemma 2.4 and Theorems 3.1 and 3.3], while the results for \(x>0\) are implied by [17, Theorems 2.2, 3.6 and 3.8]. Proposition 2 (iii) follows from [17, Theorem 3.1] for \(l=0\) and \(\mu _0=0\), and from [17, Corollary 3.2] for \(l=0\) and \(\mu _0>0\), and for \(l\ge 1\). Proposition 3 (iii) is implied by [17, Theorems 2.2, 3.3 and 3.4 (ii)], while Proposition 3 (iv) is a consequence of [17, Corollary 3.2]. Finally, the fact that \(\sigma =0\) in the setting of Proposition 1 is stated, for example, in [17, Theorem 2.2 (iv)].

### 4.2 Proof of Theorem 1

As observed already, substitution in (2) of Karlin and McGregor’s formula for \(\hat{P}_{ij}(s)\) given on [14, Equation (3.21)] leads to (3) and hence, by analytic continuation, to (16).

### 4.3 Proof of Theorem 2

Letting \(n\rightarrow \infty \) in (16) and applying the results of Propositions 1, 2 and 3 readily yields the first statement of Theorem 2.

## 5 Concluding Remarks

*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 \(C+D<\infty \) the arguments leading to Theorem 1, and hence Theorem 1 itself and Corollary 1, remain valid. Moreover, the results in [17] imply that, for \(l\ge 0\),

In the setting \(C+D<\infty \), Gong et al. [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.

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