ABSTRACT

It is well known that a supercritical single-type Bienaymé-Galton-Watson process can be viewed as a decomposable branching process formed by two subtypes of particles: those having infinite line of descent and those who have finite number of descendants. In this paper we analyze such a decomposition for the linear-fractional Bienaymé-GaltonWatson processes with countably many types. We find explicit expressions for the main characteristics of the reproduction laws for so-called skeleton and doomed particles.

1. Introduction

The Bienaymé-Galton-Watson (BGW-) process is a basic model for the stochastic dynamics of the size of a population formed by independently reproducing particles. It has a long history [1] with its origin dating back to 1837. This paper is devoted to the BGW-processes with countably many types. One of the founders of the theory of multi-type branching processes is B. A. Sevastyanov [2,3].

A single-type BGW-process is a Markov chain

 with countably many states . The evolution of the process is described by a probability generating function

(1)

wherestands for the probability that a single particle produces exactly k offspring. If particles reproduce independently with the same reproduction law (1), then the chain  represents consecutive generation sizes. In this paper, if not specified otherwise, we assume that, the branching process stems from a single particle,. Due to the reproductive independence it follows that is the n-th iteration of

.

Since zero is an absorbing state of the BGW-process, monotonely increases to a limit q called the extinction probability. The latter is implicitly determined as a minimal non-negative solution of the equation

(2)

A key characteristic of the BGW-process is the mean offspring number In the subcritical

and critical cases the process is bound to go extinct while in the supercritical case we have Clearly if and only if

In the supercritical case the number of descendants of the progenitor particle is either finite with probability q or infinite with probability Recognizing that the same is true for any particle appearing in the BGWprocess we can distinguish between skeleton particles having an infinite line of descent [4] and doomed particles having a finite line of descent. Graphically we get a picture of the genealogical tree similar to that given in Figure 1.

If we disregard the doomed particles, the skeleton particles form a BGW-process with a transformed reproduction law excluding extinction

(3)

and having the same mean. Formula (3) is usually called the Harris-Sevastyanov transformation. On the other hand, the doomed particles form another branching process corresponding to the supercritical branching process conditioned on extinction. The doomed particles produce only doomed particles according to another transformation of the reproduction law

which is usually called the dual reproduction law and has mean The supercritical BGW-process as a whole can be viewed as a decomposable branching process with two subtypes of particles [5]. Each skeleton particle must produce at least one new skeleton particle and also can give rise to a number of doomed particles. In Section 2 we describe in detail this decomposition for the single type supercritical BGW-processes.

In the special case when the reproduction generating function (1) is linear-fractional many characteristics of the BGW-process can be computed in an explicit form [6]. In Section 3 we summarize explicit results concerning decomposition of a supercritical single-type BGWprocesses.

Section 4 presents the BGW-processes with countably many types. Our focus is on the linear-fractional case recently studied in [7]. The main results of this paper are collected in Section 5 and their derivation is given in Section 6. The remarkable fact that a supercritical branching process conditioned on extinction is again a branching process was recently established in [8] in a very general setting. In general, the transformed reproduction laws are characterized in an implicit way and are difficult to analyse. This paper presents a case where the properties of the skeleton and doomed particles are very transparent.

2. Decomposition of a Supercritical Single-Type BGW-Process

The BGW-process is a time homogeneous Markov chain with transition probabilities satisfying

In the supercritical case with mean and extinction probability using the property

we can get another set of transition probabilities putting

The transformed transition probabilities also possess the branching property

where is the n-th iteration of the so-called dual generating function

The corresponding dual BGW-process is a subcritical branching process with offspring mean, see Figure 2. The dual BGW-process is distributed as the original supercritical BGW-process conditioned on extinction:

The two parts of the curve on the left panel of Figure 2 represent two transformations of the supercritical branching process. The lower-left part of the curve, replicated on the right panel of Figure 2 using a different scale, gives the generating function of the dual process. The upper-right of the curve on the left panel corresponds to the Harris-Sevastyanov transformation (3). The function (3) is the generating function for the probability distribution

with the same mean as the original offspring distribution. It is easy to see that the n-th iteration of is given by

Looking into the future of the system of reproducing particles we can distinguish between two subtypes of particles:

• skeleton particles with infinite line of descent (building the skeleton of the genealogical tree);

• doomed particles having finite line of descent.

These two subtypes form a decomposable two-type BGW-process with

The joint reproduction law for the skeleton particles has the following generating function

A check on the branching property for the decomposed process is given by

The original offspring distribution can be recovered as a mixture of the joint reproduction laws of the two subtypes

Observe also that the total number of offspring for a skeleton particle has a distribution given by

with mean. It follows,

and we can summarize the relationship among different offspring means as

3. Linear-Fractional Single-Type BGW-Process

An important example of BGW-processes is the linearfractional branching process. Its reproduction law has a linear-fractional generating function

(4)

fully characterized by two parameters: the probability of having no offspring, and the mean m of the geometric number of offspring beyond the first one. Here stands for the probability of having at least one offspring. Notice that with, the generating function (4) describes a Geometric

distribution with mean m. If the generating function (4) gives a Shifted Geometric distribution with mean If we arrive at a Bernoulli distribution.

Since the iterations of the linear-fractional function are again linear-fractional, many key characteristics of the linear-fractional BGW-processes can be computed explicitly in terms of the parameters For example, we have, and if, we get

The dual reproduction law for (4) is again linearfractional

with. The Harris-Sevastyanov transformation in the linear-fractional case corresponds to a shifted geometric distribution

Interestingly, the joint reproduction law of skeleton particles

has three independent components:

• one particle of type 1 (the infinite lineage);

• a Geometric number of offspring each choosing independently between the skeleton and doomed subtypes with probabilities and;

• a Geometric number of doomed offspring.

Observe that even though both marginal distributions and are linear-fractional, the decomposable BGW-process is not a two-type linearfractional BGW-process. The distribution of the total number of offspring for the skeleton particles is not linear-fractional

and has mean

4. BGW-Processes with Countably Many Types

A BGW-process with countably many types

describes demographic changes in a population of particles with different reproduction laws depending on the type of a particle. Here is the number of particles of type existing at generation n. In the multi-type setting we use the following vector notation:

we write if we need a column version of a vector.

A particle of type i may produce random numbers of particles of different types so that the corresponding joint reproduction laws are given by the multivariate generating functions

(5)

The offspring means

are convenient to summarize in a matrix form

.

For the n-th generation the vector of generating functions with components

are obtained as iterations of with components (5), and the matrix of means is given by The vector of extinction probabilities has its i-th component defined as the probability of extinction given that the BGW-process starts from a particle of type i. The vector is found as the minimal solution with non-negative components of equation, which is a multidimensional version of (2).

From now on we restrict our attention to the positive recurrent (with respect to the type space) case when there exists a Perron-Frobenius eigenvalue for with positive eigenvectors and such that

and

In the supercritical case, , all and we can speak about the decomposition of a supercritical BGW-process with countably many types:. Now each type is decomposed in two subtypes: either with infinite or finite line of descent. The decomposed supercritical BGW-process is again a BGW-process with countably many types whose reproduction law is given by the expressions

Linear-fractional BGW-processes with countably many types were studied recently in [7]. In this case the joint probability generating functions (5) have a restricted linear-fractional form

. (6)

The defining parameters of this branching process form a triplet, where is a sub-stochastic matrix, is a proper probability distribution, and m is a positive constant. The free term in (6) is defined as

The denominators in (6) are necessarily independent of the mother type to ensure that the iterations are also linear-fractional. This is a major restriction of the multitype linear-fractional BGW-process excluding for example decomposable branching processes.

It is shown in [7] that in the linear-fractional case the Perron-Frobenius eigenvalue, if exists, is the unique positive solution of the equation

(7)

In the positive recurrent case, when the next sum is finite

(8)

the Perron-Frobenius eigenvectors can be normalized in such a way that. They are computed as

(9)

(10)

In the supercritical positive recurrent case with and the extinction probabilities are given by

(11)

Observe that and

(12)

The total offspring number for a type i particle has mean

(13)

5. Main Results

In this section, we summarize explicit formulae that we were able to obtain for the decomposition of the supercritical linear-fractional BGW-processes with countably many types. The derivation of these results is given in the next section.

Consider the positive recurrent supercritical case with and. We demonstrate that the dual reproduction laws are again linear-fractional

(14)

with

(15)

(16)

It turns out that the following remarkably simple formulae hold for the key characteristics of the dual branching process

(17)

(18)

For the Perron-Frobenius eigenvectors we obtain the following expressions

(19)

(20)

We show that the Harris-Sevastyanov transformation results in multivariate shifted geometric distributions

(21)

where

(22)

(23)

Moreover, we demonstrate that

(24)

and

(25)

Theorem 5.1 Consider a linear-fractional BGW-process characterized by a triplet Assume it is supercritical and positively recurrent over the state space, that is and. Its dual BGW-process and its skeleton are also linear-fractional BGWprocesses with the transformed parameter triplets and with components given by Equations (15), (16), (22) and (23).

The joint offspring generating function for a skeleton particle of type has the form

(26)

where

.

Similarly to the single-type case, we can distinguish in (26) three components but now with dependence:

• a “reborn” skeleton particle of type i may change its type to j with probability;

• independent of i and j a multivariate geometric number of offspring of both subtypes;

• a linear-fractional number of doomed offspring with the fate of the first offspring being dependent on.

The total number of offspring of a skeleton particle of type i has generating function of the next form

where must belong to the interval

. The corresponding mean offspring number is larger than that given by (13):

6. Proof of Theorem 5.1

In this section we derive the formulae stated in Section 5.

Proof of (14). From

it is straightforward to obtain Equation (14) with Equations (15) and (16). We have to verify that and

The first requirement follows from (12). The second is obtained from

(27)

which is proved next. We have (relation (6) in [7])

and therefore, which is (13). Using the last two equalities and (11) we find first

and then obtain (27).

Proof of (17). In view of Equation (7) determining the Perron-Frobenius eigenvalue for a linear-fractional BGWprocess, to show (17) it is enough to verify that

Observe that according to Equation (15)

(28)

It follows,

(29)

so that we have to check that

(30)

Turning to Equation (27) we find

(31)

yielding

(32)

This and Equation (12) entail Equation (30).

Proof of (18). Starting from a counterpart of Equation (8) we find using Equation (29)

Rewrite Equation (31) as

to obtain

Thus

Proof of (19) and (20). From (15) we derive

This and a counterpart of (9)

in view of (32) brings (19)

On the other hand, a counterpart of (10) together with (28) yields

Proof of (26). We have

It follows,

Replacing the last numerator by

and dividing the whole expression by we get

and the relation (26) follows.

Proof of (21) and (24). Putting in (26) we arrive at (21). Notice that according to definition (22) and relations (12), (27) we have

Since is the unique positive solution of

and we derive

Thus and

7. Acknowledgements

Serik Sagitov was supported by the Swedish Research Council grant 621-2010-5623. Altynay Shaimerdenova was supported by the Scientific Committee of Kazakhstan’s Ministry of Education and Science, grant 0732/ GF 2012-14.

REFERENCES