Conditional Processes Induced by Birth and Death Processes

Abstract

For birth and death processes with finite state space, we consider stochastic processes induced by conditioning on hitting the right boundary point before hitting the left boundary point. We call the induced stochastic processes the conditional processes. We show that the conditional processes are again birth and death processes when the right boundary point is absorbing. On the other hand, it is shown that the conditional processes do not have Markov property and they are not birth and death processes when the right boundary point is reflecting.

1. Introduction

For one-dimensional diffusion processes on related to diffusion models in population genetics, Ewens [1] considered stochastic processes induced by conditioning on hitting the boundary point 1 before hitting the other boundary point 0. The boundary points 0 and 1 are accessible and absorbing boundaries for the diffusion processes that he considered and the induced stochastic processes are again diffusion processes. Then the induced stochastic processes are referred to as the conditional diffusion processes by Ewens [1] (see also [2]). Motivated by this work, Iizuka et al. [3] were concerned with one-dimensional generalized diffusion processes (ODGDPs for brief) on whose speed measures are right-continuous and strictly increasing functions. They considered stochastic processes induced by conditioning on hitting the right boundary point before hitting the left boundary point . The induced stochastic processes are called the conditional processes. They showed as Theorem 2.1 that the conditional processes are again ODGDPs when the boundary point is accessible with the absorbing boundary condition (Assertion 1). If the original process is a one-dimensional diffusion process with the generator then the conditional process induced by conditioning on hitting before hitting is again a one-dimensional diffusion process and its generator can be expressed as

Here we put where is a point with (see [4, 5]). On the other hand, Iizuka et al. [3] showed as Theorem 2.2 that the probability distributions of the conditional processes do not satisfy the Chapman-Kolmogorov equation when the boundary point is accessible with the reflecting boundary condition. Hence the conditional processes cannot be Markov processes when the boundary point is accessible with the reflecting boundary condition (Assertion 2).

An important class of ODGDPs which is used as stochastic models in various fields is that of birth and death processes. For example, Moran [6] introduced a birth and death process as one of fundamental stochastic models in population genetics called Moran model (we will consider this model in Section 5). However, the speed measure of any birth and death process is not a strictly increasing function (see [7]) and we cannot apply the results of [3] to birth and death processes.

In this paper we prove that Assertions 1 and 2 hold for the case that the speed measure is a nondecreasing step function. The motivation of this paper is to investigate the properties of the conditional processes induced by conditioning on hitting the right boundary point before hitting the left boundary point when the original processes are birth and death processes. The proof of Theorem 2.2 in [3] is analytical (nonprobabilistic) and it is not easy to see that the conditional processes do not satisfy Markov property when the right boundary point is accessible with the reflecting boundary condition. The proof presented in this paper for Assertion 2 is based on the fact that the state space is discrete. The proof is probabilistic and we can see intuitively that the conditional processes do not satisfy Markov property when the right boundary point is reflecting. It is our extra purpose to see this by considering birth and death processes.

In Section 2 we state our results more precisely. Section 3 is devoted to their proofs. In Section 4 we introduce a very simple birth and death process and present concrete expressions of its conditional processes considering all the boundary conditions. Finally we discuss some stochastic models in population genetics and their conditional processes in Section 5.

2. Main Results

Let be an exponentially distributed random variable with the mean 1 and let be a sequence of independent copies of . We put and (). For , an integer , and points such that , we consider a birth and death process with the state space satisfying the following conditions. For and , conditional probabilities conditional on satisfy where , and for Here denotes the probability measure concentrated at the event , that is, . The end (boundary) point [resp., ] is called to be absorbing or reflecting according to [resp., ] or [resp., ].

The generator of is given by for , where is the set of all functions on such that

Here is a proof of (5). By means of (4), we find that

Therefore we obtain the following: (see also [7]).

We show that the birth and death process can be described as an ODGDP. We set where [resp., ] is an increasing continuous function on [resp., ] such that and [resp., and ]. Further we set

Note that [resp., ] if [resp., ]. Here is a real-valued continuous increasing function on , and is a right-continuous nondecreasing function on . They are called the scale function and the speed measure, respectively. We set , , and , . We note that [resp., ] if [resp. ] is absorbing.

For a function on , we simply write [resp., ] in place of [resp., ] provided [resp., ] exists. Further, [resp., ] stands for the right [resp., left] derivative of with respect to if it exists, that is, [resp., ].

We set . Let be the space of all bounded continuous functions on satisfying the following conditions. (.1)There exist a function on and two constants such that  (.2)For each , if .

Throughout this paper we denote by an arbitrarily fixed point of . The operator is defined by the mapping from to that appeared in (11). The operator is called the one-dimensional generalized diffusion operator (ODGDO for brief) with . It is known that there exists a strong Markov process with the generator , which is called an ODGDP on (see [8, 9]). It is also known that can be identified with (see [7–9]). Indeed, it is easy to see that satisfies the following:

In order to make the boundary conditions at and clear, we use and in place of and , respectively. Here , and [resp., ] means that [resp., ] is absorbing (i.e., [resp., ]) and [resp., ] means that [resp., ] is reflecting (i.e., [resp., ]). It is known that there is the transition probability density of with respect to , that is, (see [8, 10]).

Let and let be the first hitting time at , that is, . In this paper we consider stochastic processes induced by the following conditional probability:

We set

It is known that (see [8]). We note that is independent of boundary conditions .

First we show that induces a birth and death process for .

Theorem 2.1. Assume that is absorbing. Then is independent of , and it is represented as for and . Further induces a birth and death process on for which the end point is reflecting, the end point is absorbing, and the generator is given by for , where is the set of all functions on such that

Theorem 2.1 shows that the relation between (1) and (2) for diffusion processes corresponds to the relation between (5) and (18) for birth and death processes. We note that the generator is given by (19) and the boundary condition (20) when .

Remark 2.2. By means of (9), Combining these with (18), we find that satisfies the following. For , , and , We turn to the case that is reflecting. When is strictly increasing, a representation of is given by of [3]. We note that this representation is available even if is not strictly increasing. Therefore we obtain the following representation for birth and death processes: for , , and . Here and are given as follows. For and , let For and , let It is known that and are nonnegative density functions such as (see [8]). Note that is independent of and is independent of . By virtue of [11], we see that Then we set We note that .

The second and the third terms of the right-hand side of (23) come from sample path's behavior after hitting the boundary . This representation suggests that does not satisfy Markov property. Indeed we obtain the following theorem.

Theorem 2.3. Let and . Then does not hold for some . This implies that does not satisfy Markov property.

This theorem is proved by applying the following simple proposition for sample path's behavior after hitting the boundary .

Proposition 2.4. Let and . Then does not hold for some and .

We prove this proposition in the following section.

3. Proofs of Theorems

We use the same notations as those in Section 2.

3.1. Proof of Theorem 2.1

First we prepare the following lemma. The proof of this lemma is easy and we omit it.

Lemma 3.1. Let , , and . Then it holds true that

Proof of Theorem 2.1. We assume that is absorbing. Let and . Then Let . Then by means of (15), and (26), Let . Then by using Markov property of , (15) and (32), we see that The formulas (35), (36), and (37) show that is independent of . The formula (17) follows from (13), (16), and (37).

It follows from Theorem 2.2 and Propositions 3.1 and 3.4 of [12] that induces an ODGDP on , the boundary is entrance in the sense of Feller (see [8, 13]), the boundary is absorbing, and the generator is the ODGDO with , where

Therefore, for a function on and . By means of (9), (10), and (38), we see that

In the same way, we have

Therefore we get for . Since is entrance, we see that by virtue of general theory on ODGDOs. Thus we find that is a birth and death process on , the generator is given by (18), the end point is reflecting with (19), and the end point is absorbing. The proof is completed.

3.2. Proof of Theorem 2.3

We introduce the Green function corresponding to . For , , and , let be a continuous function on satisfying the following properties:

Here . It is known that there exist such functions (see [8]). We set . Note that is a positive number independent of . We put for and , which is the Green function corresponding to . It is also known that for and (see [8, 14]).

First we prove Proposition 2.4.

Proof of Proposition 2.4. We divide the proof into four cases.

Case 1. . Since is absorbing, we find that by means of (34). Since there are such that for , (54) shows that (31) does not hold true for for .
Let Then which imply that (31) is not valid for and . Thus (31) does not hold for some .

Case 2. and . By means of (33), we also see that The right-hand side of this formula is negative if . This implies that (31) does not hold true for .

Case 3. and . By means of (34), Since is absorbing, we get Combining these equalities with (33) and (34), we see that The right-hand side of this formula is positive if . This implies that (31) does not hold true for .

Case 4. . Suppose that (31) holds true for , and . Then for , and , where By means of (34), for and .

Combining this with (53), we see that for and , where stands for the expectation with respect to . Here we note that (64) is valid for . Indeed,

Combining this with (53), we see that which implies (64) with .

Since (64) holds true for and , we have for . By virtue of (48), (51), and (52), we see that

We take a point such that . Then by virtue of (50), (51), and (52),

Thus we obtain that

It is known that (see [8, 9, 14, 15]). Therefore letting in (70) leads us to This contradicts the fact that the last term is positive. Thus (31) does not hold true for , , and .

Remark 3.2. Let , , and . Then we see that where . Therefore that is, (31) is valid for . Proposition 2.4 implies, however, that (31) does not hold for and .

Proof of Theorem 2.3. Let , , and . Then, by using Markov property of , we obtain that

Therefore (30) is equivalent to the following:

Again (76) is equivalent to the following:

Since , (77) is equivalent to

However (78) does not hold true for some and by virtue of Proposition 2.4. Thus (30) does not hold true for some and .

4. Examples

In this section, we consider a simple birth and death process. Let , , and , where . For given in Section 2, the transition law of this birth and death process satisfies that

4.1. Case That the End Point 1 Is Absorbing

We first consider , that is,

Then and . Further (9) and (10) are reduced to

By virtue of [11], we obtain that where [resp., ] if [resp., ]. Further by virtue of Theorem 2.1, for , where . Note that is expressed by using . As in Remark 2.2, this induces a birth and death process on with the transition law

4.2. Case That the End Point 0 Is Absorbing and the End Point 3 Is Reflecting

We next consider with , that is,

For simplicity, we put . Then and . Further (9) and (10) are reduced to

By virtue of [11], we obtain that

By means of (23), we obtain that for , where is given by (82) with , and

4.3. Case That the End Points 0 and 3 Are Reflecting

We finally consider with and , that is,

For simplicity, we put . Then and . Further (9) and (10) are reduced to

By virtue of [11], we obtain that

By means of (23), we obtain that

for , where is given by (82) with , and and are given by (89). Further is given as follows:

5. Conditional Processes in Population Genetics

Here we consider two stochastic models in population genetics and their conditional processes. In this section, we use notations different from those of the previous sections to emphasize the difference between the original models and the induced models of conditional processes. We denote the conditional process by [resp., ] when the original process is [resp., ] as we did in Section 1.

5.1. Diffusion Model

We consider the following diffusion model for a randomly mating population consisting of haploid individuals with two types (alleles) and . Let be the relative frequency of at time . Then is a one-dimensional diffusion process on with the generator where [resp., ] is mutation rate from [resp., ] to [resp., ] (see [4]).

First we consider the case that . The point 1 is accessible and exit boundary if (see [16]). For this diffusion process, consider a stochastic process induced by conditioning on hitting the boundary point 1 before hitting the other boundary point 0. The induced stochastic process is again a diffusion process with the generator by (2) and Theorem 2.1 of [3]. Note that the effect of conditioning is that it inflates the mutation rate to . Ewens [1] considered the case that and the induced diffusion process is referred to as the conditional diffusion process by Ewens [1] (see also [2]).

Next we consider the case that . The point 1 is regular boundary in this case (see [16]) and we can pose various boundary conditions there. If we pose the absorbing boundary condition, then the induced process is again a diffusion process with the generator by (2) and Theorem 2.1 of [3]. On the other hand, if we pose the reflecting boundary condition as it is usually done in population genetics (see [17–19]), then the induced conditional process does not satisfy the Chapman-Kolmogorov equation and this process is not a diffusion process due to Theorem 2.2 of [3]. These results imply that we cannot use the diffusion model whose generator is given by (97) as the conditional process when we pose the reflecting boundary condition at the boundary point 1.

5.2. Moran Model

Moran [6] introduced the following birth and death process as one of the fundamental stochastic models in population genetics called continuous-time Moran model (see [4] for discrete time Moran model). We refer to this model as Moran model for brief. Let be the number of individuals in a haploid population with two types and , where is an integer greater than 2. Let and , be a sequence of random times introduced in Section 2. At time an individual is chosen randomly and it reproduces a new individual (). The type of the newborn individual is [resp., ] with probability [resp., ] and it is [resp., ] with probability [resp., ] if the parent is [resp., ], where . Then at this time an individual except newborn individual is chosen randomly to die. There is no change at time (). Denoting by the relative frequency of at time , is a birth and death process on with the transition law where (). Note that , , , , , and . Note also that unless and , unless and , and for . The process does not jump at time if the types of newborn individual and the dead are the same even though a “birth and death” event occurs at time . One of the end points 0 is absorbing [resp., reflecting] when [resp., ] and the other end point 1 is absorbing [resp., reflecting] when [resp., ]. Let be the first hitting time to ().

First we consider the case that . This is the case without mutation and both boundary points are absorbing with

By (9) and (10) we have

Then Theorem 2.1 implies that the conditional process conditional on is again a birth and death process on with the transition law where and for . The end point is reflecting since . Note that of the original Moran model reduces to if and this is essentially the same as the simple birth and death process discussed in Section 4.1.

Next we consider the case that . The boundary point 1 is reflecting. Then the induced conditional process does not satisfy Markov property and this conditional process is not a birth and death process by Theorem 2.3.

6. Acknowledgments

The authors thank Thomas Nagylaki for suggesting them to consider conditional processes for Moran model in population genetics. They also thank the anonymous reviewer for comments on the previous version of this paper.