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As the classic branching process, the Galton-Watson process has obtained intensive attentions in the past decades. However, this model has two idealized assumptions–discrete states and time-homogeneity. In the present paper, we consider a branching process with continuous states, and for any given , the branching law of every particle in generation n is determined by the population size of generation We consider the case that the process is extinct with Probability 1 since in this case the process will be substantially different from the size-dependent branching process with discrete states. We give the extinction rate in the sense of and almost surely by the form of harmonic moments, that is to say, we show how fast grows under a group of sufficient conditions. From the result of the present paper, we observe that the extinction rate will be determined by an asymptotic behavior of the mean of the branching law. The results obtained in this paper have the more superiority than the counterpart from the existing literature.
Branching process is an important class of Markov processes, which describes the survival and extinction of a particle system. The most classical branching process is called the Galton-Watson process (see ). For a chosen family, Galton and Watson  used this process to record the number of males in each generation. For a Galton-Watson process , we usually set , which means that there is a male ancestor in the family. The relationship between and is written by
where presents the number of boys whose father (in generation n) is indexed by i. In a Galton-Watson process, the random array is set to be i.i.d. Hence, Galton-Watson process is a time homogeneous Markov chains with discrete state. There are two idealized assumptions in this model: one is the discrete state space, the other is the property of time homogeneous. In other words, there are two directions to extend this model.
Jiřina process (see [2,3,4,5]) is the continuous version of the Galton-Watson process, which stresses that the role of can take value in () instead of Since the state space of this process is a subset of , we use the Laplace transform to describe the relationship between the number of particles in generation n and , which is described by
where is a cumulate generate function of a certain infinitely divisible distribution G. G can be observed as the common branching mechanism (i.e., the law of ) of each particle. It should be noted that in the above equality, is independent of n, thus, we see that the Jiřina process is still time-homogeneous.
To break the feature of time homogeneous, several time-inhomogeneous branching processes have been studied over the past decades. There are different motivations to construct the time-inhomogeneous property for a branching process, one of which assumes that the common law of is depending on , and takes value in for every . We call this a time-inhomogeneous branching process as the size-dependent branching process (with discrete time and discrete state). This assumption (the law of depends on ) has a strong practical background; for example, when a country is overpopulated, the government may promote family planning, while if a country faces the problem of population scarcity, the government will encourage childbearing. This model has been investigated in [6,7,8] and some other papers.
In the present paper, the model we consider is the continuous version of the size-dependent branching process, which is also called the generalized Jiřina process (for short, GJP). This model was introduced in , where the model is defined by the Laplace transform as
where is called a reproduction cumulative function (for short, r.c.f.) and it has the following representation:
We can refer to  on how to obtain (2). On the other hand, ref.  also explains that is a non-negative Borel function, and is a bounded kernel from to . That is to say,
Hence, we see that the r.c.f. is determined by and Obviously, if there exist a constant r and a measure v on such that
then GJP will degenerate to the Jiřina process. Moreover, from (1) one can see
Actually, we have set that is a bounded kernel. Denote
then we have
which means that presents the expectation of the children reproduced by unit parent when the generation of the parent contains x particle(s). The above equality is equivalent to
By a direct calculation we obtain
For a branching process , a very important topic, which is usually considered first, is the limit behavior of and the distribution of the limit (if it exists). For example, the celebrated Kesten-Stigum theorem (see , Chapter 1) for the Galton-Watson process and various generalized Kesten-Stigum theorem for different types of branching processes (see [3,7,10]). In summary, the Kesten-Stigum theorem and its various of generalized versions demonstrate that converges to 0 with Probability and to with Probability and depends on the branching mechanism (reproduction law) of the branching process. Ref.  showed that the asymptotic behavior of GJP also behaves as
and is depending on some properties of . The author of  also pointed out that it is as similar as the asymptotic behavior of size-dependent branching process for the case . The most interesting and worth investigating is the case that , since when the state space is , then means that there exists a finite generation n such that but can always be positive even though when the state space is . Under some mild assumptions, ref.  gave the extinction rate of in the sense of almost surely when . The idea to deal with the extinction rate is to consider the growth rate of , then, the method to show the growth rate of the size-dependent branching process when can be referred. Ref.  gave a sufficient condition to ensure that the extinction rate in the sense of it almost surely is also the extinction rate in the sense of . In the present paper, we obtain a new extinction rate, which is easier to understanding by the definition of the mean function (see Section 3 for detail). Combining with the result in , we can observe that an extinct GJP may have different extinction rates under different conditions.
In this paper, we consider the rate of in the sense of almost surely and when the GJP behaves as . We will give another group of sufficient conditions to ensure that there exists a constant sequence such that has a limit in the sense of almost surely and . Compared with the previous results, our results have more values for applications.
The GJP has a strong connection with reality. We can use GJP to model a number of chemical reactions and biological situations. For instance, it is proper to describe the trend of the concentration by GJP for some bacteria or virus whose reproduction depend on their concentration in the medium. For more examples, we recommend  and the references therein.
2. Main Results
For the sake of presenting our results, first of all, we give some basic assumptions as follows.
There exits a function for all which satisfies that and
is non-increasing, is non-decreasing and concave, and
For any , it satisfies .
is non-decreasing, concave and is concave.
For any , it satisfies .
We remind that if exists on then (A2) implies (A4). Denote
First, we give some lemmas and results which will be used during, as we prove our main theorems.
Suppose thatis a positive and non-increasing function, then for any, the following propositions are equivalent:
According to the conclusion in Theorem 1 we obtain . Hence, ones have
That is to say, we arrive at
From Lemma 1 we have , which means that
and thus the limit exists. Now, we construct a martingale as
Denote as the -norm of the random variable X, hence, it is clear that
It is obvious that for any n, one has
Moreover, from the estimate in the proof of Theorem 3, we have
Since is a concave function (see Assumption (A4)), we can obtain that
Since , then it follows that
Thus, by utilizing Lemma 1, it is not hard to verify that
which establishes that
Combining the above inequality with the fact that is a martingale, we claim that has a limit in the sense of from the martingale convergence theorem. On the other hand, we observe that also has the limit since is a Cauchy sequence. Recall that and we have shown that has the limit S in the sense of almost surely, then we have
Moreover, , thus, . That is to say, S is non-degenerate. □
Compared with the results in , the assumptions in the present paper do not need that . We also even do not require that . Intuitively, will make the process more likely to be extinct. Hence, is not a natural enough condition under the case , which we consider. Moreover, the extinction rate may be different between in  and in this paper, since under the assumption in  the rate will be (if it exists). One can see that there are many cases (for example, the case that is not depending on x) in which . We remind that the rate in our paper appears reasonable because of , and further, we consider the case that the process is extinct.
Throughout our paper, under the Assumptions (A1)–(A5), we obtain an extinction rate for a GJP in the sense of almost surely and , which enriches the limit theory of GJP process. Therefore, our results have potential values in applications.
Y.L. designed the research, wrote the paper and gave the support of funding acquisition. H.H. made some revisions to the paper. All authors have read and agreed to the published version of the manuscript.
The first author acknowledges the Fundamental Research Funds for the Central Universities (No. 2232021D-30). The second author acknowledges the financial support from the Natural Science Foundation of Chongqing of China (No. cstc2020jcyj-msxmX0762), and the Initial Funding of Scientific Research for High-level Talents of Chongqing Three Gorges University of China (No. 2104/09926601).
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
The data presented in this study are available upon request from the corresponding author.
The authors thank the editor and the referees for their valuable comments and suggestions, which improved greatly the quality of this paper.
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