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Branching Processes in Varying Environment

1.1. Introduction

Branching processes are a fundamental object in probability theory. They serve as models for the reproduction of particles or individuals within a collective or a population. Here we act on the assumption that the population evolves within clearly distinguishable generations, which allows us to examine the population at the founding generation n = 0 and the subsequent generations n = 1, 2, . To begin with, we focus on the sequence of population sizes Zn at generation n, n = 0. Later, we shall study whole family trees.

Various kinds of randomness can be incorporated into such branching models. For this monograph, we have two such types in mind. On the one hand, we take randomness in reproduction into account. Here a main assumption is that different individuals give birth independently and that their offspring distributions coincide within each generation. On the other hand, we consider environmental stochasticity. This means that these offspring distributions may change at random from one generation to the next. A fundamental question concerns which one of the two random components will dominate and determine primarily the model's long-term behavior. We shall get to know the considerable influence of environmental fluctuations.

This first chapter is of a preliminary nature. Here we look at branching models with reduced randomness. We allow that the offspring distributions vary among the generations but as a start in a deterministic fashion. So to speak we consider the above model conditioned by its environment.

We begin with introducing some notation. Let (0) be the space of all probability measures on the natural numbers 0 = {0, 1, 2, .}. For f ? (0), we denote its weights by f[z], z = 0, 1,.. We also define

The resulting function on the interval [0, 1] is the generating function of the measure f. Thus, we take the liberty here to denote the measure and its generating function by one and the same symbol f. This is not just as probability measures and generating functions uniquely determine each other but operations on probability measures are often most conveniently expressed by means of their generating functions. Therefore, for two probability measures f1 and f2, the expressions f1f2 or f1 f2 do not only stand for the product or composition of their generating functions but also stand for the respective operations with the associated probability measures (in the first case, it is the convolution of f1 and f2). Similarly, the derivative f´ of the function f may be considered as well as the measure with weights f´[z] = (z + 1)f[z + 1] (which in general is no longer a probability measure). This slight abuse of notation will cause no confusions but on the contrary will facilitate presentation. Recall that the mean and the normalized second factorial moment,

can be obtained from the generating functions as

NOTE.- Any operation we shall apply to probability measures (and more generally to finite measures) on 0 has to be understood as an operation applied to their generating functions.

We are now ready for first notions. Let (O, , P) be the underlying probability space.

DEFINITION 1.1.- A sequence v = (f1, f2, .) of probability measures on 0 is called a varying environment.

DEFINITION 1.2.- Let v = (fn, n = 1) be a varying environment. Then a stochastic process = {Zn, n ? 0} with values in 0 is called a branching process with environment v, if for any integers z = 0, n = 1

On the right-hand side, we have the Zn-1th power of fn. In particular, Zn = 0 P-a.s. on the event that Zn-1 = 0. If we want to emphasize that probabilities P(·) are determined on the basis of the varying environment v, we use the notation Pv(·).

In probabilistic terms, the definition says, for n = 1, that given Z0, . Zn-1 the random variable Zn may be realized as the sum of i.i.d. random variables Yi,n, i = 1, ., Zn-1, with distribution fn,

This corresponds to the following conception of the process : Zn is the number of individuals of some population in generation n, where all individuals reproduce independently of each other and of Z0, and where fn is the distribution of the number Yn of offspring of an individual in generation n - 1. The distribution of Z0, which is the initial distribution of the population, may be arbitrary. Mostly we choose it to be Z0 = 1.

EXAMPLE 1.1.- A branching process with the constant environment f = f1 = f2 = · · · is called a Galton-Watson process with offspring distribution f. ?

The distribution of Zn is conveniently expressed via composing generating functions. For probability measures f1, . fn on 0 and for natural numbers 0 = m < n, we introduce the probability measures

Moreover, let fn,n be the Dirac measure d1.

PROPOSITION 1.1.- Let be a branching process with initial size Z0 = 1 a.s. and varying environment (fn, n = 1). Then for n = 0, the distribution of Zn is equal to the measure f0,n.

PROOF.- Induction on n. ?

Usually it is not straightforward to evaluate f0,n explicitly. The following example contains an exceptional case of particular interest.

EXAMPLE 1.2.- LINEAR FRACTIONAL DISTRIBUTIONS. A probability measure f on 0 is said to be of the linear fractional type, if there are real numbers p, a with 0 < p < 1 and 0 = a = 1, such that

with q = 1 - p. For a > 0, this implies

We shall see that it is convenient to use the parameters and instead of a and p. Special cases are, for a = 1, the geometric distribution g with success probability p and, for a = 0, the Dirac measure d0 at point 0. In fact, f is a mixture of both, i.e. f = ag + (1 - a)d0. A random variable Z with values in 0 has a linear fractional distribution, if

that is, if its conditional distribution, given Z = 1, is geometric with success probability p. Then

For the generating function, we find

(leading to the naming of the linear fractional). It is convenient to convert it for > 0 into

Note that this identity uniquely characterizes the linear fractional measure f with mean and normalized second factorial moment .

The last equation now allows us to determine the composition f0,n of linear fractional probability measures fk with parameters From f0,n = f1 f1,n,

Iterating this formula we obtain (with 1 · · · k-1 := 1 for k = 1)

It implies that the measure f0,n itself is of the linear fractional type with a mean and normalized second factorial moment

This property of perpetuation is specific for probability measures of the linear fractional type. ?

For further investigations, we now rule out some cases of less significance.

ASSUMPTION V1.- The varying environment (f1, f2, .) fulfills 0 < n < 8 for all n = 1.

Note that, in the case of n = 0, the population will a.s. be completely wiped out in generation n.

From Proposition 1.1, we obtain formulas for moments of Zn in a standard manner. Taking derivatives by means of Leibniz's rule and induction, we have, for 0 = m < n,

and . In addition, using the product rule, we obtain after some rearrangements

and Evaluating these equations for m = 0 and s = 1, we get the following formulas for means and normalized second factorial moments of Zn, which we had already come across in the case of linear fractional distributions (now the second factorial moments may well take the value 8).

PROPOSITION 1.2.- For a branching process with initial size Z0 = 1 a.s. and environment (f1, f2, .) fulfilling V 1, we have

We note that these equations entail the similarly built formula

set up for the standardized variances

of the probability measures fk. Indeed,

1.2. Extinction...