Critical Galton-Watson processes with overlapping generations

A properly scaled critical Galton-Watson process converges to a continuous state critical branching process $\xi(\cdot)$ 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 $(\int_0^y\xi(y-u)du^\gamma, y\ge0)$ with a pertinent $\gamma\ge0$.


Introduction
One of the basic stochastic population model of a self-reproducing system is build upon two assumptions Within this model, the numbers of individuals Z 0 , Z 1 , . . ., born at times t = 0, 1, . . ., form a Markov chain, whose transition probabilities are fully described by the distribution of the offspring number N . The Markov chain {Z t , t ≥ 0} is usually called a Galton-Watson process, or GW-process for short. A GW-process is classified as subcritical, critical, or supercritical, depending on whether the mean offspring number E(N ) is less than, equal to, or larger than the critical value 1.
It is known, that in the critical case, with E(N ) = 1, Var(N ) = 2b, b < ∞, (1.1) the finite dimensional distributions (fdd's) of a properly scaled GW-process converge 2) and the limiting fdd's are represented by a continuous state branching process ξ(·), 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 is the variance of N , the faster is changing the population size.
In this paper, we study {Z(t), t ≥ 0}, a Galton-Watson process with overlapping generations, or GWO-process for short, where Z(t) is the number of individuals alive at time t in a reproduction system satisfying the following two assumptions Assumption (B * ) allows for overlapping generations, when mothers may coexist with their daughters. We focus on the critical case (1.1) and aim at an extension of (1.2) to the GWO-processes. The process {Z(t), t ≥ 0}, being non-Markov in general, is studied with help of an associated renewal process, introduced in Section 2. The mean inter-arrival time a := E(τ 1 + . . . + τ N ) (1.5) 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 µ := E(L), 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 Z t and the number of individual alive Z(t) at the time t. (For GW-processes, a = 1 and Z(t) equals Z t , since all alive individuals are newborn.) An interesting case of population counts is treated by Theorem 4 dealing with decomposable multitype GWprocesses. 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 our Theorems 1, 2, and 3 respectively. In these theorems, it is always assume that the GWO-process stems from a large number Z 0 = n of progenitors born at time zero. Corollary 1. Consider a GWO-process satisfying (1.1) and a < ∞. If µ < ∞, then Corollary 2. Consider a GWO-process satisfying (1.1) and a < ∞. If µ = ∞, and for some slowly varying function at infinity L(·), t j=0 P(L > j) = t γ L(t), 0 ≤ γ ≤ 1, t → ∞, (1.6) then, as n → ∞, Corollary 3. Consider a GWO-process satisfying (1.1), a < ∞, and (1.6). Then, as n → ∞, Notice that condition (1.6) holds even in the case µ < ∞, with γ = 0 and L(t) → µ as t → ∞. The family of processes {ξ γ (·)} γ≥0 emerging in our limit theorems can be expressed in the integral form ξ 0 (u) := ξ(u), for γ = 0, and ξ γ (u) : which is treated as a convenient representation of the limiting fdd's, see Section 4.
The following remarks comment on relevant literature and mention an interesting open problem.
1. The GW-process is a basic model of the biologically motivated theory of branching processes, see [1], [6]. The critical GW-process can be viewed as a stochastic model of a sustainable reproduction, when a mother produces on average one daughter, see [11].
2. We distinguish between a stronger and a weaker forms of the uniform convergence which respectively require the relations to hold for any 0 < y 0 < y 1 < ∞.
3. We will write E n (·) := E(· | Z 0 = n) 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 ξ(·).
4. We will often use the following two shortenings Note that both these functions are increasing, and for 0 ≤ x ≤ y, (1.9) 5. In different formulas, the symbols C, C 1 , C 2 , c, c 1 , c 2 represent different positive constants.

Population counts
The number of individuals alive at time t can be counted as the sum of individual scores where L jk is the life length of an individual born at time j, and χ jk (t) = 1 {0≤t<L jk } is its individual score. In this case, 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.
Definition 2.1 For a progenitor of the GWO-process, define its individual score as a vector (χ(t)) t∈Z with non-negative, possibly dependent components, such that χ(t) = 0 for all t < 0. 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 X(t) = X [χ] (t) as the sum of time shifted individual scores assuming that the individual scores (χ jk (t)) t∈Z are independent copies of (χ(t)) t∈Z .

The litter sizes
In terms of (1.4), the litter sizes of a generic individual are defined by ν(t) := N j=1 1 {τj=t} , t ≥ 1, so that ν(1) + . . . + ν(L) = N . On the other hand, given a random infinite dimensional vector where ν(t) is treated the litter size at age t for an individual with the life length L, the consecutive ages at childbearing can be found as where N (t) is the number of daughters produced by a mother of age t.
Observe that the corresponding mean inter-arrival time is indeed given by (1.5):

Associated renewal process
In the GWO setting with Z 0 = 1, the process Z t conditioned on {N (t) = k}, can be viewed as the sum of k independent daughter copies Z t = Z (1) t−τ k . This branching property implies that the expected number of newborns U (t) := E 1 (Z t ) satisfies a recursive relation where the * symbol stands for a discrete convolution Resolving the obtained recursion U (t) = 1 {t=0} + U * A(t), 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 W t , the waiting time to the next renewal event R t (j) := P(W t = j), j ≥ 1, t ≥ 0.
These probabilities satisfy the renewal equation R t (j) = A(t + j) + R t * A(t), which yields By the key renewal theorem, there exists a stable distribution of the residual time W t , in that If r(y) → r(0) as y → 0, then using the assumptions on r (n) (·) and r(·). It remains to notice that → 0 as first t → ∞ and then t 0 → ∞.

Expected population counts
If Z 0 = 1, then X(t), defined by (2.1), can be represented as in terms of the independent daughter processes X (j) (·). Taking expectations, we arrive at a recursion where M (t) := E 1 (X(t)), m(t) := E(χ(t)). This renewal equation and applying the key renewal theorem, we conclude The obtained parameter m χ can be viewed as the average χ-score for the population with overlapping generations. The next result goes further than (2.8) by giving a useful asymptotical relation in the case m χ = ∞.
Proof We have to show that (2.9) implies M (t) − a −1 M t = o(M t ) as t → ∞, where M t := t j=0 m(j). To this end, observe that the difference for an arbitrarily small ǫ > 0 and some finite constants C, t ǫ . It remains to apply the property of the regularly varying function M t , saying that M t − M t−c = o(M t ) as t → ∞ for any fixed c ≥ 0.
Turning to X(t) = Z(t), the number of individuals alive at time t, observe that with In this case, the parameter m χ = µa −1 can be treated as the degree of generation overlap. For example, m χ = 2 means that on average, the life length L covers two generation lengths.

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 χ(t) > 0 for t < 0. 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.
The purpose of this section is to introduce a branching renewal equation for Λ(·) and establish Proposition 3.5, which will play a key role in the proofs of the main results of this paper.
Lemma 3.2 For a given vector (t 1 , . . . , t p ) with non-negative integer components, consider the log-Laplace transform of the p-dimensional distribution of the population sum X(·) defined by (2.1). Then, in accordance with Definition 3.1, Proof It suffices to observe that

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 (f (t)) t∈Z , see more on it in Section 3.2. The convolution term Ψ[Λ] * U (t) represents the non-linear part of the branching renewal equation. A seemingly free term B(·) of the equation (3.2) is a non-negative function specified below by (3.4) and (3.6). It also depends on the function Λ(·) 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 (2.7) where X (j) (·) are independent daughter copies of (X(·)|Z 0 = 1). It entails e χ(t)−X(t) = N j=1 e −X (j) (t−τj) , and taking expectations, we obtain On the other hand (recall e x 1 : Denoting the last expectation D(t), we can write 4) due to independence between the progenitor score χ(t) 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 R t (j) is given by (2.5). Here we used , we conclude that relation (3.2) holds with

Laplace transform of the reproduction law
The Laplace transform of the reproduction law E e −f (τ1)−...−f (τN ) is a positive functional defined on the set of non-negative sequences (f (t)) t≥1 . The higher than first moments of the joint distribution of (τ 1 , . . . , τ N ) are characterised by the non-linear functional This functional is non-negative and monotone in view of the elementary equality Proof The first assertion follows from the relation connecting Ψ[f ](t) and Ψ(f ). The second assertion follows from the L'Hospital rule. Proof Observe, that (3.8) implies which in turn gives for arbitrary 1 ≤ t 1 ≤ t, where f := sup t≥1 | f (t) | and Using E(N (N − 1)) = 2b, we therefore obtain This implies that Applying (3.9) with t 1 = nǫ, t = ny, and Thus, under the imposed conditions, for any y 0 > 0. It remains to observe that n 2 Ψ[z n ](ny) y → br 2 (y) as n → ∞, according to Lemma 3.3.

Basic convergence result
If Λ(t) is given by Definition 3.1, then This observation explains the importance of the next result.
If the non-negative functions B n (t) are such that where B(y) is a continuous function, then where r(y) is a continuous function uniquely defined by Proof. We will prove this statement in three steps. Firstly, we will show where δ n (y) stands for a function (different in different formulas) such that δ n (y) y → 0 as n → ∞. Secondly, putting ∆ n (y) := nΛ n (ny) − r(y), we will find a y * > 0 such that Thirdly, we will demonstrate that ∆ n (y) Proof of (3.14). Rewriting (3.13) as r(y) = B(y) − b y 0 r 2 (y − u)a −1 du, and using (2.4), (3.12), we obtain r(y) = nB n (ny) − bn −1 ny t=0 r 2 (y − tn −1 )U (t) + δ n (y). This and Lemma 3.4 imply (3.14).

Laplace transforms for ξ(·)
The set of functions with u i , λ i ≥ 0, determines the fdd's for the process ξ(·). Proof This result is obtained by induction, using (1.3) and the Markov property of ξ(·). To illustrate the argument, take p = 2 and non-negative y, y 1 , y 2 . We have With u 2 = y 2 and u 1 = y 1 + y 2 , this gives an explicit expression for the asserted relation E e −λ1ξ(u1+y)−λ2ξ(u2+y) | ξ(y) = x = e −xG2(ū,λ) in the case p = 2. then for all y ≥ 0, assuming G 0 (ū,λ) := 0, the following two relations hold Proof With u p = 0, relation (4.1) gives Then the function H (n) (·) satisfies a recursion Proof Putting f k := F ( k n ) and f −1 := 0, we get and by Lemma 4.1, with u i := k−i n and λ i = f i−1 − f i−2 for i ≥ 1. Since by Lemma 4.2, we conclude Proof Applying a Taylor expansion to the recursion of Lemma 4.3, we obtain .
By reiterating this recursion, we get To prove the lemma, it suffices to verify that where H (n) (·) satisfies (4.6), with f i = F ( i n ). To this end, note that implies an upper bound where δ n (ny) y ⇒ 0 as n → ∞. In view of this relation, we can find a sufficiently small y * > 0, such that sup 0≤y≤y * |∆ n (ny)| → 0, n → ∞.

Laplace transforms for ξ • F (·)
Notice that the Riemann-Stieltjes integrals appearing in this paper are understood as Proof The proof of Lemma 4.5 uses similar argument as Lemma 4.3 and Proposition 4.4, with the main idea being to demonstrate that the step function version of (4.8), defined by converges H (n) p (ū + y,λ) → H p (ū + y,λ) to the solution of (4.9) as n → ∞. Instead of giving tedious details in terms of the discrete version of (4.8), we indicate below the key new argument in terms of continuous version of the integral ξ • F (·).
Due to (4.8), we have which in view of (1.3) and (4.8), yields

Splitting each of the integrals in two parts
and then, using the Markov property of the process ξ(·) we obtain where F p (y) := H p−1 (ū,λ) + F • p (y). After this, it remains to apply Proposition 4.4.

Main results
The aim of this chapter is to establish an fdd-convergence result for the vector (X 1 (·), . . . , X q (·)) composed of the population counts corresponding to different individual scores χ 1 (·), . . . , χ q (·), which may depend on each other.
There are three new features in the limiting process of (5.1) compared to that of (1.2): the continuous time parameter u does not include zero, reflecting the fact that it may take some time for the distribution of ages of coexisting individuals to stabilise, the time scale a −1 corresponds to the scaling by the average length of overlapping generations, the factor m χ accounts for the average χ-score in a population with overlapping generations.
Theorem 2 Consider a population count defined by (2.1). Assume (1.1), a < ∞, (2.9), and in the case Then where ξ γ (·) is given by (1.7), which is understood according to the previous chapter.
The next result extends Theorems 1 and 2 to the case of several population counts.
Theorem 3 Consider q ≥ 1 population counts X 1 (t), . . . , X q (t), each defined by Definition 2.1 in terms of different individual scores χ 1 (t), . . . , χ q (t). Assume (1.1), a < ∞, and (2.9), with γ = γ j and L = L j for the χ j -score, j = 1, . . . , q. If m χj = ∞, assume additionally condition (5.2) for the χ j -score. Then, as n → ∞, To illustrate the utility of Theorem 3, we consider a multitype GW-process t is the number of type i-individuals born at time t, for i = 1, . . . , q. Each individual of type i is assumed to live one unit of time and then be replaced by N ij individuals of type j. Denoting m ij := E(N ij ), assume that the multitype GW-process is decomposable in that The next result deals with a decomposable critical GW-process, satisfying To put this process into the GWO-framework, we treat as GWO-individuals only the type 1 individuals, while the other types will be addressed by respective population counts. Clearly, the numbers of GWO-individuals forms a single type GW-process, and (1.2), derived from Corollary 1, describes the limit behaviour of the scaled process (Z 1 t , t ≥ 0|Z 1 0 = n). Since the process {Z 1 0 , . . . , Z 1 n−1 | Z 1 0 = n} during n units of time, produces type 2 individuals, of order n new individuals per unit of time, one would expect, in view of Theorem 3, a typical number of type 2 individuals at time n to be of order n 2 . An extrapolation of this reasoning suggests scaling by n j for the number of type j individuals, j = 1, . . . , q.
Here the limiting process ξ(·) is the same as in (1.2) and ξ j (y) = y 0 ξ(y − u)du j . Notice that the only source of randomness in the q-dimensional limit process is due to the randomly fluctuating number of the first type of individuals. Observe also, that only the means m j,j+1 appear in the limit, but not the other means like for example m 1,3 . This fact reflects the following phenomenon of the reproduction system under consideration: in a large population, the number of type 3 individuals stemming directly from type 1 individuals is much smaller compared to the number of type 3 individuals stemming from type 2 individuals.
Turning to (5.22), we split its left-hand side in three parts using (1.9), and then produce an upper bound as a sum of three terms involving an arbitrary k ≥ 1: The third term is handled by (5.24). The first term is further estimated from above by where the right-hand side converges to zero for any fixed k. Finally, in view of the proof of (5.22) is finished by applying Fatou's lemma as k → ∞.

The last relation follows from the upper bound
because the third term tends to 0 as n → ∞, thanks to (5.20), and the first two terms in the right-hand side vanish as k → ∞ and k 1 → ∞ due to the assumption m χ < ∞.

Proof of Theorem 2
The main idea of the proof of Theorem 2 is the same as of Theorem 1, and here we mainly focus on the new argument addressing the case m χ = ∞. We want to prove (5.7) with the modified annotations where H p (ū,λ) is defined by (4.8), with F (y) := y γ . In this case, relation (5.9) holds with ψ n,p (t) := p i=1 λ n,i χ(nu i + t), λ n,i := λ i n −1−γ L −1 (n), and according to (4.9), the right-hand side of (5.7) satisfies Thus, under the conditions of Theorem 2, relation (5.7) will follow from Proposition 3.5 after we show nB n (ny) where B n (t) is defined by (5.13) and (5.14). Its counterpart (5.12) was proven in the case m χ < ∞ according to flow chart (5.8). In the rest of the proof, we follow the same flow chart and comment on necessary changes in the case m χ = ∞. The counterparts of (5.17). (5.20), and (5.18) in the case m χ = ∞, are verified in a similar way as in the case m χ < ∞, now using Proposition 2.3. The counterpart of (5.19) takes the form as Proposition 2.3 yields the following counterpart of (5.21) To verify (5.22) in the case m χ = ∞, we check that after applying the Cauchy-Schwartz inequality for expectations To prove the counterpart of (5.23) in the case m χ = ∞, we use a sequence of upper bounds E(ψ n,p (t)) < ǫC(y 1 ), n ≥ 1, 0 ≤ y ≤ y 1 holds for an arbitrary ǫ > 0. Sending ǫ → 0 ends the proof of (5.23) and thereby of Theorem 2.

Proof of Theorem 3
Lemma 5.1 Put Then for u 1 > . . . > u p = 0, the following integral equation holds Proof The lemma is proven similarly to Lemma 4.5. According to (5.28), the limit function in (5.7) satisfies the integral equation Therefore, to apply Proposition 3.5, we have to prove for the updated version of (5.13), that nB n (ny) y → r p−1 (0) + a −1 F p,q (y), n → ∞, which once again, is done according to flow chart (5.8). Even in this more general setting, the counterparts of (5.17) and (5.18) are valid, and the task boils down to verifying the counterpart of (5.19) where the limit is obtained using Proposition 2.3 for the counterpart of (5.21) It remains to verify the counterparts of (5.22), (5.23).
Using (1.9), we can split the left-hand side of (5.22) into the sum of three terms ).
The first and the second terms are handled using the argument of the proofs of Theorems 1 and 2 respectively.
The third term requires a special attention. It is estimated from above by E(ψ ′ n,p−1 (t)).
The last term is tackled in a similar way as (5.24), and it remains to show that for each j ≤ s ′ , ny t=1 E χ j (t)e ψ ′′ n,p (t) 1 y ⇒ 0, n → ∞.
The first term is taken care by (5.24), while the second term vanishes as k → ∞ since m χj < ∞.
Proof of (5.23). Using ψ n,p (t) = ψ ′ n,p (t) + ψ ′′ n,p (t), we get e ψn,p(t) 1 ≤ e ψ ′ n,p (t) 1 + e ψ ′′ n,p (t) 1 , which allows us to replace (5.23) by the following two relations The first relation is proven in the same way as (5.23) was proven for Theorem 1, and the second relation is proven in the same way as (5.23) was proven for Theorem 2.