A properly scaled critical Galton–Watson process converges to a continuous state critical branching process as the number of initial individuals tends to infinity. We extend this classical result by allowing for overlapping generations and considering a wide class of population counts. The main result of the paper establishes a convergence of the finite-dimensional distributions for a scaled vector of multiple population counts. The set of the limiting distributions is conveniently represented in terms of integrals ( , ) with a pertinent .
One of the basic stochastic population models of a self-reproducing system is built upon the following two assumptions:
different individuals live independently from each other according to the same individual life law described in (B);
an individual dies at age one and at the moment of death gives birth to a random number 𝑁 of offspring.
It is known that, in the critical case, with
the finite-dimensional distributions (fdds) of a properly scaled GW-process converge,
and the limiting fdds are represented by a continuous state branching process , which is a continuous time Markov process with a transition law determined by
Note how the parameter 𝑏 acts as a time scale: the larger the variance of 𝑁, the faster the change of the population size.
In this paper, we study , a Galton–Watson process with overlapping generations, or GWO-process for short, where is the number of individuals alive at time 𝑡 in a reproduction system satisfying the following two assumptions:
different individuals live independently from each other according to the same individual life law described in (B*);
an individual lives 𝐿 units of time and gives 𝑁 births at random ages , satisfying(1.4)
The process , being non-Markov in general, is studied with help of an associated renewal process, introduced in Section 2. The mean inter-arrival time
of this renewal process gives us the average generation length. It is important to distinguish between the average generation length 𝑎, which in this paper will be assumed finite, and the average life length , allowed to be infinite.
With a more sophisticated reproduction mechanism (1.4), there are many interesting population counts to study, alongside the number of newborns and the number of individuals alive at the time 𝑡. Observe that is the total number of daughters produced at time 𝑡 by individuals alive at time . In particular, in the GW setting, and since all alive individuals are newborn.
An interesting case of population counts is treated by Theorem 4 dealing with decomposable multitype GW-processes. Theorem 4 is obtained as an application of the main results of the paper, Theorems 1, 2, 3, stated and proven in Section 5. The following three statements are straightforward corollaries of Theorems 1, 2, and 3 respectively. In these theorems, it is always assumed that the GWO-process stems from a large number of progenitors born at time zero.
Consider a GWO-process satisfying (1.1) and . If , then
Consider a GWO-process satisfying (1.1) and . If and, for some slowly varying function at infinity ,
then, as ,
Notice that condition (1.6) holds even in the case , with and as . The family of processes emerging in our limit theorems can be expressed in the integral form
which is treated as a convenient representation of the limiting fdds; see Section 4.
The following remarks comment on relevant literature and mention an interesting open problem.
The GW-process is a basic model of the biologically motivated theory of branching processes; see [1, 7, 17]. The critical GW-process can be viewed as a stochastic model of a sustainable reproduction, when a mother produces on average one daughter; see . On the convergence result (1.2) for the critical GW-processes, see [1, 2, 10, 13].
Our limit theorems are stated in terms of the fdd-convergence. Finding simple conditions on the individual scores, ensuring weak convergence in the Skorokhod sense, is an open problem.
To avoid confusion, we set apart discrete and continuous variables:
Mixed products are treated as integer numbers so that stands for . The latter results in not always being equal to 𝑢.
We distinguish between a stronger and a weaker forms of the uniform convergence
which respectively require the relations
to hold for any .
We will write
to say that the expected value is computed under the assumption that the GWO-process starts from 𝑛 individuals born at time 0. With a little risk of confusion, we will also write
when the expectation deals with the finite-dimensional distributions of the continuous state branching process .
We will often use the following two shortenings:
Note that both these functions are increasing, and for ,(1.8)(1.9)
In different formulas, the symbols represent different positive constants.
2 Population Counts
The number of individuals alive at time 𝑡 can be counted as the sum of individual scores
where is the life length of the 𝑘-th individual born at time 𝑗 (according to an arbitrary labelling of the individuals born at time 𝑗) and is its individual score. Here the individual score is 1 if the individual is alive at time 𝑡, and 0 otherwise. This representation leads to the next definition of a population count.
For a progenitor of the GWO-process, define its individual score as a vector with non-negative, possibly dependent components such that for all . This random vector is allowed to depend on the individual characteristics (1.4), but it is assumed to be independent from such characteristics of other individuals.
Define a population count as the sum of time shifted individual scores
assuming that the individual scores are independent copies of .
2.1 The Litter Sizes
In terms of (1.4), the litter sizes of a generic individual are defined by , , so that . On the other hand, given the random infinite-dimensional vector
where is treated as the litter size at age 𝑡 for an individual with the life length 𝐿, the consecutive ages at childbearing can be found as
where is the number of daughters produced by a mother of age 𝑡.
In the critical case, the probabilities
sum up to one since . A renewal process with inter-arrival times having distribution plays a crucial role in the analysis of the critical GWO-processes. Observe that the corresponding mean inter-arrival time is indeed given by (1.5),
In order to avoid a possible confusion, we emphasise at this point that and if .
2.2 Associated Renewal Process
In the GWO setting with , the process conditioned on , where is the birth count of the founder, can be viewed as the sum of 𝑘 independent daughter copies . This branching property implies that the expected number of newborns satisfies a recursive relation
where the ∗ symbol stands for a discrete convolution
Resolving the obtained recursion , we find a familiar expression for the renewal function
so that, by the elementary renewal theorem,
This says that, in the long run, the underlying reproduction process produces one birth per 𝑎 units of time. In this sense, 𝑎 can be treated as the average generation length.
Later on, we will need the following facts concerning the distribution of , the waiting time to the next renewal event:
These probabilities satisfy the renewal equation , which yields
By the key renewal theorem, there exists a stable distribution of the residual time , in that
Assume (1.1), , and suppose a family of non-negative functions is such that
If as , then
for any . From
using the assumptions on and . It remains to notice that
as first and then . ∎
2.3 Expected Population Counts
If , then , defined by (2.1), can be represented as
in terms of the independent daughter processes , where is the birth count of the founder. Taking expectations, we arrive at a recursion
where , . This renewal equation yields
and applying the key renewal theorem, we conclude
The obtained parameter can be viewed as the average 𝜒-score for the population with overlapping generations. The next result goes further than (2.5) by giving a useful asymptotic relation in the case .
Consider a critical GWO-process with . If, for some function slowly varying at infinity,
then as .
We have to show that (2.6) implies as , where . To this end, observe that the difference
is estimated from above by
for an arbitrarily small and some finite constants 𝐶, . It remains to apply the property of the regularly varying function , saying that as for any fixed . ∎
Turning to , the number of individuals alive at time 𝑡, observe that, with ,
Therefore, given ,
In this case, the parameter can be treated as the degree of generation overlap. For example, means that, on average, the life length 𝐿 covers two generation lengths.
3 Branching Renewal Equations
A useful extension of Definition 2.1 broadens the range of individual scores by replacing (2.1) with
Relation (3.1) takes into account even those individuals who are born after time 𝑡, allowing for . In this paper, we refer to this extension only to deal with the finite-dimensional distributions of the population counts defined by (2.1); see Lemma 3.2 below.
For the population count given by (3.1), the log-Laplace transform is given by
The purpose of this section is to introduce a branching renewal equation for and establish Proposition 3.5, which will play a key role in the proofs of the main results of this paper.
For a given vector with non-negative integer components, consider the log-Laplace transform
of the 𝑝-dimensional distribution of the population sum defined by (2.1). Then, in accordance with Definition 3.1,
It suffices to observe that
3.1 Derivation of the Branching Renewal Equation
Here we show that Definition 3.1 leads to what we call a branching renewal equation,
where the operator
is defined on the set of non-negative sequences ; see more on it in Section 3.2. The convolution term represents the non-linear part of the branching renewal equation. A seemingly free term of equation (3.2) is a non-negative function specified below by (3.4) and (3.5). It also depends on the function in a non-linear way; however, asymptotically it acts as a truly free term.
The derivation of (3.2) is based on the following extended version of decomposition (2.4):
where are independent daughter copies of . It entails , and taking expectations, we obtain
On the other hand (recall ),
Denoting the last expectation , we can write
due to independence between the progenitor score and the GWO-processes stemming from progenitor’s daughters. Combing the previous relations, we find
which, after introducing a term involving operator (3.3), brings
Subtracting both sides from 1 yields
which can be rewritten in the form of a renewal equation
Formally solving this renewal function, we get
where is given by (2.3). Here we used
Since , we conclude that relation (3.2) holds with
3.2 Laplace Transform of the Reproduction Law
The Laplace transform of the reproduction law is a positive functional defined on the set of non-negative sequences . The higher than first moments of the joint distribution of are characterised by the non-linear functional
This functional is monotone in view of the elementary equality
in that if for all , then . In particular, with , we get due to our standing assumption , which implies that for all eligible .
The earlier introduced operator (3.3) is obtained from functional (3.6) through the connection
which is verified by
Consider a constant function , . If (1.1), then
and as .
The first assertion follows from the relation connecting and . The second assertion follows from the L’Hospital rule. ∎
If (1.1) holds and
where is a continuous function, then
3.3 Basic Convergence Result
If is given by Definition 3.1, then
This observation explains the importance of the next result.
Assume (1.1), , and consider a sequence of positive functions satisfying
If the non-negative functions are such that
where is a continuous function, then
where is a continuous function uniquely defined by
We will prove this statement in three steps. Firstly, we will show
where stands for a function (different in different formulas) such that as . Secondly, putting , we will find a such that
Thirdly, we will demonstrate that
Proof of (3.13). Rewriting (3.12) as
and using (2.2), (3.11), we obtain
This and Lemma 3.4 imply (3.13).
Proof of (3.14). Relations (3.10) and (3.13) yield
Under the current assumptions, the inequality implies that the sequence of functions is uniformly bounded over any finite interval . Therefore, putting into (3.8) gives
for any fixed . Combining this with (3.16) entails
so that, for some positive constant independent of ,
It follows that
Replacing here 𝑣 by , we derive
which, after letting , results in (3.14).
Proof of (3.15). It suffices to demonstrate that the convergence interval in (3.14) can be consecutively expanded from to , from to , and so forth. Suppose we have established that, for some ,
Then, for , by (3.17),
Since , we may conclude that
thereby completing the proof of (3.15). ∎
4 Continuous State Critical Branching Process
In this section, among other things, we clarify the meaning of given by (1.7), in terms of the log-Laplace transforms of the fdds of the process . From now on, we consistently use the following shortenings:
4.1 Laplace Transforms for
The set of functions
with , determines the fdds for the process .
For non-negative ,
This result is obtained by induction, using (1.3) and the Markov property of . To illustrate the argument, take and non-negative . We have
With and , this gives an explicit expression
for the asserted relation