#### Abstract

Let be a supercritical continuous-time branching process with immigration; our focus is on the large deviation rates of and thus extending the results of the discrete-time Galtonâ€“Watson process to the continuous-time case. Firstly, we prove that is a submartingale and converges to a random variable . Then, we study the decay rates of and for and under various moment conditions on and . We conclude that the rates are supergeometric under the assumption of finite moment generation functions.

#### 1. Introduction

Suppose that is a continuous-time branching process with immigration. Its generating matrix is defined as follows:where

Throughout this paper, we always suppose that , , and .

Setwhere and separately represent the total number of the original individualâ€™s offspring and the offspring of the individual from immigrants at the time . If for all and , then the process degenerates to the continuous-time branching process . It is known from Athreya and Ney [1] or Harris [2] that is a martingale and converges to a r.v. W w.p.1 as . Therefore, converges to w.p.1 as for arbitrary fixed , and henceFor , definewhere and .

In this paper, let , we will demonstrate that is a submartingale and converges to a r.v. almost surely since converges to a r.v. a.s. and converges to a r.v. a.s.

The above discussion of the decay rates is not only of its own meaning but also of some significance to the algorithm tree structure in computer science (Karp and Zhang [3] and Miller [4]). Heyde [5] discussed that there exists a constant sequence which makes converge to a random variable W w.p.1. Athreya [6] considered similar decay rates for the Galtonâ€“Watson process. Seneta [7] researched the supercritical Galtonâ€“Watson process with immigration, and Riotershtein [8] considered multitype branching processes with immigration in a random environment. Liu and Zhang [9] studied the decay rates of for the supercritical branching processes with immigration. Recently, Sun and Zhang [10] considered the convergence rates of harmonic moments for supercritical branching processes with immigration. Chen and He [11] investigated the lower deviation and moderate deviation probabilities for maximum of a branching random walk. Abraham et al. [12] analyzed the stationary continuous state branching processes. The model considered in this paper involves continuous-time and immigration, while the models considered in the above references do not involve continuous-time. Such change in conditions may affect the large deviation rates and thus is mathematically interesting. We need to find a new method to investigate the effect of continuous-time and immigration.

Based on the previous results, it is natural to develop the large deviation rates for the continuous-time branching processes with immigration. In this paper, we are aiming to discuss the decay rates ofunder various moment conditions on and for any and . We conclude that the rates of (6) are always supergeometric under the assumption of finite moment generation functions. The specific conclusions are presented in Section 3.

#### 2. Preliminaries

Before the investigation, we present the generating functions and of the known sequences and as

Clearly, is well defined and finite at least on with , so 1 is the solution of . Then, is the mean birth rate of . Moreover, is similar to .

We present some preliminaries in this section.

Lemma 1. *Let and . Ifexists, then for arbitrary ,*

*Proof. *Omitted.

Lemma 2. * exists for any , and . Furthermore, satisfies the functional equation:with condition .*

*Proof. *By the Kolmogorov forward equation,For ,That is,Hence,For ,So,By Lemma 1,It follows from (11) thatHence,Finally, we can obtain the following equality from (11):and moreover,Taking the limit as , we haveThe proof is completed.

Proposition 1. *Let be the -algebra generated by and . Then, is a submartingale and converges to a r.v. almost surely.*

*Proof. *According to the definition of , we conclude the following inequalities:which implies is a submartingale. We know that if and only if . Hence, . Thus, is an integrable submartingale and converges to a r.v. almost surely.

Define , with initial condition , where is a continuous-time branching processes with immigration with . Denote andWe will study the properties of and in the subsequent contents. Let . Apparently, is strictly increasing with and . Furthermore, we have for since and for since . Therefore, the iterates of are nonincreasing with and nondecreasing with (with respect to ).

Proposition 2. *Let . If for some , then for , as andwhere is the unique solution ofsubject to*

*Proof. *Note that for , we know that for , . Hence, is well defined on for and for .

Since , we have for any , which implying as . On the other hand,that is,Denote . Putting yields . Thus, we obtain , i.e., as .

Now, there exists for such thatHence, exists.

Furthermore, satisfies (26) and (27). For ,Letting yieldsFor another,since for . Thus, .

Finally, it is easily seen that the solution of (26) and (27) is unique.

Proposition 3. *Let for some . Then,where is the round of .*

*Proof. *For , noting yieldsAssume that inequality (35) holds for , i.e.,Then,holds for all .

Therefore, (35) is proved.

#### 3. Main Results and Proofs

Theorem 1. *Assume that and for some . Then, for arbitrary small ,where .*

*Proof. *Note that could be defined aswhere and are independent and identical distributed, are i.i.d. random variables with the same law as . Hence,Indeed, for any fixed ,where and are arbitrary constants. Thus,where .

Since it is assumed that and , we have and it follows that there exist and such thatfor any and . Hence, there exist such that for all . By Markovâ€™s inequality, we can prove that there exist and such that . Therefore, by Lemma 2,The proof is completed.

Theorem 2. *Assume that for fixed and , there exist constants and such that and for all . Then, (39) holds.*

*Proof. *Under the above assumptions, there exists another such that for all since is equivalent to as . In general, we denote the positive constants by . Therefore,With a simple modification of the convergence theorem, it can be proved thatsinceFor all ,and henceSince , the proof shall be completed if we provewhere . Indeed, for any fixed , let be the inverse function of at , then we have as . Note that , thus we can writewhereSince , it follows promptly that satisfyingBy hypothesis and hence for any , we can pick an such that for all . Hence, for ,which impliesand henceApplying for together with (57) implies (51).

Corollary 1. *Assume that for some and such that . Then, (39) holds.*

*Proof. *By Markovâ€™s inequality, we haveAccording to the assumption,is finite, and then there exists s.t. for all . Applying Theorem 2, the proof is completed.

Theorem 3. *Assume that for some and . Then, there exists satisfying*

*Proof. *In general, we suppose that for . Firstly, we prove thatSince is equivalent to , we haveHence, (61) is proved since . By Proposition 3, note that for any ,Indeed, if , by Proposition 3,But, by definition of ,and hence the above inequality becomessince . Hence, as since . Finally, by Proposition 2,which is positive and finite. Therefore, we can find such that

Theorem 4. *Assume that for some and . Then, there exist constants and satisfying*