Decomposition of Supercritical Linear-Fractional Branching Processes ()
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)
where
stands 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)
Figure 1. An example of a BGW-tree up to level
. Solid lines represent the infinite lines of descent and dotted lines represent the finite lines of descent.
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:
Figure 2. Duality between the subcritical and supercritical cases. Left: a supercritical generating function (1) with two positive roots 
for the Equation (2). Right: the dual generating function drawn on a different scale.
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.