On a stochastic model for continuous mass branching population

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Abstract

It is well known that the set of non-negative integers is the state-space of usual branching stochastic processes. However in many applications one may have situations when it is difficult to count the number of individuals in the population, but some non-negative characteristic, such as volume, weight or product produced by the individuals can be measured. To model this kind of situation, branching stochastic processes with continuous state-space are introduced. In this paper two theorems which establish relationship between asymptotic behavior of processes continuous and discrete state-space and with immigration in varying environment will be proved.

Introduction

The theory of branching stochastic processes is a rapidly developing part of the general theory of stochastic processes. During quite a long time the main object of investigation in the theory of branching processes was the number of individuals (particles) at a given time. So in classic models of branching processes the state-space is the set of non-negative integers. However in many applications one may have situations when it is difficult to count the number of individuals in the population, but some non-negative characteristic, such as volume, weight or product produced by the individuals can be measured. At the end of sixties Jirina [5], [6] defined a branching stochastic process with continuous state space as a homogeneous Markov process the transition probabilities of which satisfy some “branching condition”. Later many papers were published to study this kind of processes (see Refs. [4], [8], [9], [11], [12]). A model of the branching process with continuous state space has appeared in [10] as limiting for branching processes with generalized immigration. Kallenberg [7] introduced a branching model with continuous state-space and studied it under the assumption that the “offspring distribution” is infinitely divisible. Adke and Gadag [1] defined a new model of continuous state-space process with immigration using in the “branching condition” a counting process with independent and stationary increments.

In the paper [1] the authors investigated some distributional properties (and the extinction probability) of the process in the case when the offspring and immigration distributions vary from generation to generation (so-called varying environments). However the asymptotic results are obtained under the assumption of fixed environments. On the other hand in applications the immigration rate may usually be affected by seasonal and global changes of the environment. In this paper we will study asymptotic behavior of the continuous state-space branching stochastic process with immigration in varying environments.

It is convenient to define the process, which we are going to consider, as a family of non-negative random variables describing the amount of a product produced by individuals of the population. The initial state of the process is given by a non-negative random variable X0. The amount of the product X1 of the first generation is defined as the sum of random products produced by N1(X0) individuals and the product U1 of immigrating to the first generation individuals. Similarly the amount X2 of the product of the second generation is defined as the sum of products produced by N2(X1) individuals and U2, and so on. Here Nk(t),k1,tT, are counting processes with independent stationary increments, T is either R+=[0,) or Z+={0,1,2,} and Uk,k1, are non-negative random variables. In the paper [1] limit distributions are obtained for process Xn when Un,n1 are i.i.d. random variables, which corresponds to the fixed environment. In particular when the process is critical it is shown that the linearly normalized process has a gamma limiting distribution. If one does not assume that Un,n1 have a common distribution, several rather important questions appear: (i) under which conditions we may still use the linear normalization to get a non-degenerate limiting distribution? (ii) When will the limiting gamma distribution be preserved? (iii) How does a change of the rate of immigration affect the asymptotic behavior of the process? In this paper we expect to obtain results which give answers for these kinds of questions.

Section snippets

Main theorems

We now give a detailed definition of the process which we are going to consider. Let {Win,i,n1} be a double array of independent and identically distributed non-negative random variables, {Nn(t),tT,n1} be a family of non-negative, integer-valued independent processes with independent stationary increments, with Nn(0)=0 almost surely, T is either R+=[0,) or Z+={0,1,}.

We define a new process Xn,n0, as following. Let the initial state of the process be X0 which is an arbitrary non-negative

The Foster–Williamson theorem

As it was indicated before process Z(n)=Nn(Xn1) is a Galton–Watson process with immigration. The offspring distribution and the distribution of the number of immigrating “individuals” have Laplace transforms G(f(λ))=Eeλξn and Hn(f(λ))=Eeληn, respectively (see [1]). Here ξn=Nn(Wn1),ηn=Nn(Un1) and f(λ)=logEeλNn(1).

We obtain the moments of offspring distribution by standard arguments as following:m=Eξn=ddλG(f(λ))λ=0=EWEN,where N=N1(1),W=W1 and for B=Eξn(ξn1) we obtain:B=EW[varNEN]+EW2(EN)

Acknowledgments

These results are part of the project no. FT-2005/01 funded by KFUPM, Dhahran, Saudi Arabia. I am indebted to King Fahd University of Petroleum and Minerals for excellent research facilities.

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