A critical branching process with immigration in varying environments

doi:10.1016/j.spl.2020.108928

Statistics & Probability Letters

Volume 168, January 2021, 108928

https://doi.org/10.1016/j.spl.2020.108928 Get rights and content

Abstract

The paper studies a critical Bienaymé–Galton–Watson branching processes with immigration in varying environments. Assuming that the offspring variance is infinite and the mean number of immigrants is either finite or infinite is proved the asymptotic formulas for the probability for non extinction. The proper limiting distributions under the appropriate normalization are also proved.

Introduction

A branching process is said to be in varying environments if the offspring distribution of the particles changes with time. Most of the classical branching processes have been studied also in varying environments. See e.g. D’Souza, 1994, Cohn and Jagers, 1994, Grey and Zhunwei (1994), Mitov et al. (2003), Mitov and Yanev (2002), Batra and Gupta (2005), Yu and Xu (2010), González et al. (2019) and Kersting (2020) and the books (Athreya and Ney, 1972, Jagers, 1974).

The investigation of a critical Bienaymé–Galton–Watson (BGW) branching processes with infinite variance in varying environments was initiated in Mitov and Yanev (2002). In the paper (Mitov and Omey, 2014) the authors studied the limiting behavior of a critical BGW process when the offspring mean equals unit, but the variance increases to infinity. Here it is assumed that the variance is infinite, and simultaneously the environment varies from generation to generation. The immigration component also changes the behavior of any branching process. By this reason we consider two regimes of immigration: mean number of immigrants in each generation is either finite or infinite. Let us mention also that both varying environments and the immigration make the BGW branching processes more flexible and enlarge the possible applications of these processes for modeling in biology, cell proliferation, finance etc.

The paper is organized as follows. Section 2 contains the definitions and the basic conditions. The functional equations for the process are derived also in this section. Section 3 contains the asymptotic of the probability for non-extinction. Limit theorems are proved in Section 4. Section 5 contains some concluding remarks.

Section snippets

Definitions and basic equations

The Bienaymé–Galton–Watson branching process ${Z_{n}}$ can be defined as follows $Z_{0} = 1, Z_{n + 1} = I_{(Z_{n} > 0)} \sum_{i = 1}^{Z_{n}} X_{i} (n + 1), n = 0, 1, 2, \dots$ , where $X_{i} (n), i = 1, 2, \dots; n = 0, 1, 2, \dots$ are independent, non negative integer valued random variables. If these random variables are identically distributed for every fixed $n$ but the distribution varies from generation to generation, the process is said to be in varying environments.

Let ${I_{n} : n = 1, 2, \dots}$ be independent identically distributed (iid) integer valued random variables, independent

Probability for non-extinction

In this section we prove the asymptotic formulas for the probability $Pr (Y_{n} > 0)$ under the basic condition (1) and two different regimes of immigration. The following lemma is needed.

Lemma 1

Let $A_{n} = \sum_{j = 1}^{k (n)} {(a (n) + j)}^{- s}$ , where $s > 1$ , $a (n) > 0$ for all $n = 1, 2, \dots$ , and $k (n) \to \infty, a (n) \to a \geq 0$ , as $n \to \infty$ . Then $lim_{n \to \infty} A_{n} = \sum_{j = 0}^{\infty} {(a + 1 + j)}^{- s} = ζ (s, a + 1),$ where $ζ (s, q) = \sum_{j = 0}^{\infty} {(q + j)}^{- s}$ is the Hurwitz zeta function with $s, q \in C, Re (s) > 1, Re (q) > 0$ .

Remark 1

Notice that $ζ (s) = ζ (s, 1) = \sum_{j = 1}^{\infty} j^{- s}$ is the Riemann $ζ$ function.

Proof

Let $ε \in (0, 1)$ be fixed. Then, for all $n \geq n (ε)$ one

Limit theorems

Theorem 3

Assume that (1) holds and the first moment of the number of immigrants is finite, i.e. $m_{I} = g^{'} (1) < \infty$ . Then for $x \geq 0$ $Pr (Y_{n} c {(n)}^{- 1 ∕ θ} \leq x | Y_{n} > 0) \to D (x), n \to \infty,$ where $D (x)$ has Laplace transform $\hat{D} (λ) = 1 - ζ (1 ∕ θ, λ^{- θ} + 1) ∕ ζ (1 ∕ θ)$ .

Proof

It is known that for $λ > 0$ $E [e^{- λ Y_{n} c {(n)}^{- 1 ∕ θ}} | Y_{n} > 0] = 1 - \frac{1 - Φ (n; e^{- λ c {(n)}^{- 1 ∕ θ}})}{1 - Φ (n; 0)} .$

Since the asymptotic of $1 - Φ (n; 0)$ is studied in Theorem 1, now we have to investigate the asymptotic of $1 - Φ (n; e^{- λ c {(n)}^{- 1 ∕ θ}})$ for fixed $λ > 0$ . Let us denote $s (n) = e^{- λ c {(n)}^{- 1 ∕ θ}}$ . From (2) we get $log Φ (n; s (n)) = \sum_{j = 1}^{n} log g (1 - ({(1 - s (n))}^{- θ} +))$

Concluding remarks

A unified approach to the branching processes in varying environment is given in Kersting (2020). Since the mean of the offspring distribution changes with time, the classification of the processes as subcritical, critical or supercritical differs form the classical one. In the present paper Eq. (1) provides that the mean of the offspring distribution equals 1, so that the classical definition of criticality is valid and it is used.

In the recent paper (González et al., 2019) the authors studied

Acknowledgments

The author wishes to thank the reviewers for their valuable suggestions and comments which improved the paper significantly.

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At the very beginning, there is only a basic particle, and after a random time period, the particle splits into multiple independent particles that experience the same activities. Different versions of branching processes have been developed to describe various particle reproduction processes and branching paths between two generations of particles [10–13]. So far, branching processes have been used to describe the spread of epidemic diseases, transport of contaminants in groundwater, and fatigue crack growth, etc.
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^☆: The work is partially supported by the NFSR of the Min. Edu. Sci. of Bulgaria , Grant No. KP-06-H22/3.

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A critical branching process with immigration in varying environments☆

Abstract

Introduction

Section snippets

Definitions and basic equations

Probability for non-extinction

Limit theorems

Concluding remarks

Acknowledgments

Stat. Probab. Lett.

Branching Processes

A branching process in varying environments: A mixed mating model

J. Ind. Soc. Agril. Stat.

Regular Vriation

General branching processes in varying environments

Ann. Appl. Prob.

The rates of growth of the Galton-Watson process in varying environments

Advances in Applied Probability

Branching processes in varying environment with generation dependent immigration

Stoch. Models