Galton-Watson theta-processes in a varying environment

We consider a special class of Galton-Watson theta-processes in a varying environment fully defined by four parameters, with two of them $(\theta,r)$ being fixed over time $n$, and the other two $(a_n,c_n)$ characterizing the altering reproduction laws. We establish a sequence of transparent limit theorems for the theta-processes with possibly defective reproduction laws. These results may serve as a stepping stone towards incisive general results for the Galton-Watson processes in a varying environment.


Introduction
The basic version of the Galton-Watson process (GW-process) was conceived as a stochastic model of the population growth or extinction of a single species of individuals [3,7].The GW-process {Z n } n≥0 unfolds in the discrete time setting, with Z n standing for the population size at the generation n under the assumption that each individual is replaced by a random number of offspring.It is assumed that the offspring numbers are independent random variables having the same distribution {p(j)} j≥0 .
By allowing the offspring number distribution {p n (j)} j≥0 to depend on the generation number n, we arrive at the GW-process in a varying environment [4].This more flexible model is fully described by a sequence of probability generating functions Introduce the composition of generating functions Given that the GW-process starts at time zero with a single individual, we get E(s Zn ) = F n (s), P(Z n = 0) = F n (0).
The state 0 of the GW-process is absorbing and the extinction probability for the modeled population is determined by q = lim F n (0) (here and throughout, all limits are taken as n → ∞, unless otherwise specified).In the case of proper reproduction laws with f n (1) = 1 for all n ≥ 1, we get .
In [5], the usual ternary classification of the GW-processes into supercritical, critical, and subcritical processes [1], was adapted to the framework of the varying environment.Given 0 < f ′ n (1) < ∞ for all n, it was shown that under a regularity condition (A) in [5], it makes sense to distinguish among four classes of the GW-processes in a varying environment: supercritical, asymptotically degenerate, critical, and subcritical processes.In a more recent paper [10] devoted to the Markov theta-branching processes in a varying environment, the quaternary classification of [5] was further refined into a quinary classification, which can be adapted to the discrete time setting as follows: supercritical case: q < 1 and lim E(Z n ) = ∞, asymptotically degenerate case: q < 1 and lim inf E(Z n ) < ∞, strictly subcritical case: q = 1 and a finite lim E(Z n |Z n > 0) exists, loosely subcritical case: q = 1 and lim E(Z n |Z n > 0) does not exist.
Our paper is build upon the properties of a special parametric family of generating functions [9] leading to what will be called here the Galton-Watson theta-processes or GW θ -processes.The remarkable property of the GW θ -processes in a varying environment is that the generating functions F n (s) have explicit expressions presented in Section 2. An important feature of the GW θ -processes is that they allow for defective reproduction laws.If the generating function f i (s) is defective, in that f i (1) < 1, then F n (1) < 1 for all n ≥ i.In the defective case [6,11], a single individual, with probability 1 − f i (1) may force the entire GW-process to visit to an ancillary absorbing state ∆ by the observation time n with probability In Sections 3 and 4, we state ten limit theorems for the GW θ -processes in a varying environment.These results are illuminated in Section 5 by ten examples describing different growth and extinction patterns under environmental variation.The proofs are collected in Section 6.

Proper and defective reproduction laws
A GW θ -process with parameters (θ, r, a n , c n ) n≥1 is a GW-process in a varying environment characterized by a sequence of probability generating functions (f n (s)) n≥1 defined by for θ = 0, and for θ = 0, defined by Definition 1 is motivated by the Definitions 14.1 and 14.2 in [9], which also mentions a trivial case of θ = −1 not included here.Observe that in the setting of varying environment, the key parameters θ ∈ (−1, 1] and r ≥ 1, stay constant over time, while the parameters (a n , c n ) may vary.The case θ = r = 1 is the well studied case of the linear-fractional reproduction law.This section contains two key lemmas.Lemma 1 gives the explicit expressions for the generating functions Lemmas 2 presents the asymptotic properties of the constants A n , C n , D n leading to the limit theorems stated in Sections 3 and 4.
Lemma 1.Consider a GW θ -process with parameters (θ, r, a n , c n ).If θ = 0, then Here, 3 Limit theorems for the proper GW θ -processes Theorems 1, 2, 3, 4, 5 deal with the GW θ -process in the case θ ∈ (0, 1], r = 1, when by Lemma 1, These five theorems fully cover the five regimes of reproduction in a varying environment and could be summarized as follows.Let θ loosely subcritical if the lim B n does not exist, see Theorem 5.
This section also includes Theorem 6 addressing the proper case θ = 0, r = 1.Notice that Theorem 6 deals with the case of infinite mean values, when the above mentioned quinary classification does not apply.
and Z n almost surely converges to a random variable Z ∞ such that and with Theorem 5. Let θ ∈ (0, 1], r = 1, and assume that lim B n does not exist.Then q = 1 and letting kn , and with λ n = λB Theorem 6. Suppose θ = 0 and r = 1.Then P(Z n > 0) = D n , so that q = 1 − D, with D given by Lemma 2(e).Furthermore, (i) if A = 0 and D = 0, then q = 1 and if A = 0 and D > 0, then q < 1 and ) and D = 0, then q = 1 and (iv) if A ∈ (0, 1) and D > 0, then q < 1 and Z n almost surely converges to a random variable

Remarks
1.It is a straightforward exercise to check that the above mentioned regularity condition (A) in [5] is valid for the GW θ -process in the case θ ∈ (0, 1], r = 1.
2. The limiting distribution obtained in Theorem 3 coincides with that of [12] obtained for the critical GW-processes in a constant environment with a possibly infinite variance for the offspring number.
3. The statement (ii) Theorem 6 is of the Darling-Seneta type limit theorem obtained in [2] for GWprocesses with infinite mean.
4. Part (iv) of Theorem 6 presents the pattern of limit behavior similar to the asymptotically degenerate regime in the case of infinite mean values.The conditions of Theorem 6 (iv) hold if and only if (1 4 Limit theorems for the defective GW θ -process In the defective case, there are two kinds of absorption times: τ 0 the absorption time of the GW θ -process at 0, τ ∆ the absorption time of the GW θ -process at the state ∆. Let τ = min(τ 0 , τ ∆ ) be the absorption time of the GW θ -process either at 0 or at the state ∆.Recall that q = P(τ 0 < ∞) and denote Clearly, P(τ Theorems 7-10 present the transparent asymptotical results on these absorption probabilities and the limit behavior of the GW θ -process in the four defective cases.Corollaries of Theorems 7-9 deal with the proper sub-cases, where τ = τ 0 .All three corollaries describe a strictly subcritical case, when A = 0, and an asymptotically degenerate case, when A ∈ (0, 1).Theorem 7. Consider the case θ ∈ (0, 1], r > 1.Then so that Q = 1.Furthermore, and Z n almost surely converges to a random variable Z ∞ taking values in the set {∆, 0, 1, 2, . ..}, with Corollary.Consider the case θ ∈ (0, 1], r > 1 assuming so that C = (1 − A)(r − 1) −θ implying q ∆ = 0.

Proofs
In this section we sketch the proofs of lemmas and theorems of this paper.The corollaries to Theorems 7-9 are easily obtained from the corresponding theorems.

Proof of Lemma 2
(a) In the case θ ∈ (0, 1], r = 1, the claim follows from the existence of lim C n and lim(A n + C n ), which in turn, follows from monotonicity of the two sequences.To see that The

Church-Lindvall condition for the GW θ -process
In [8] it was shown for the GW-processes in a varying environment that the almost surely convergence Z n a.s.
Proof.In view of (13), we have It remains to observe that given A ∈ (0, 1) the relation D ∈ (0, 1) is equivalent to Lemma 5. Assume that θ = 0 and r > 1, and consider { Zn }, a GW-process in a varying environment with the proper probability generating functions Proof.Assume θ ∈ (0, 1] and r > 1 together with (3).Then A n → A ∈ (0, 1), a n → 1, and c n → 0. We have are such that h n ≥ 1 and k n ∈ (0, 1].The statement follows from the representation where H = n≥1 h n and K = n≥1 k n .It is easy to show that (3) and ( 1 On the other hand, K ∈ (0, 1], since In the other case, when (3) holds together with θ ∈ (−1, 0) and r > 1 , the lemma is proven similarly.
Proof of Theorems 1, 2, 3, 4, 5 The proofs of these theorems are done using the usual for these kind of results arguments applied to the explicit expressions available for F n (s).In particular, the following standard formula is a starting point for computing the conditional limit distributions Thus in the case θ ∈ (0, 1] and r > 1, Lemma 1 and ( 16) imply proving the main statement of Theorem 4. The almost sure convergence stated in Theorem 2 follows from Lemma 3 and the earlier cited criterium of [8].In other words, P(Z n ≤ we x/An |Z n > 0) → (1 − e −x )1 {0≤w<∞} .

entailing the main claims of Lemma 1 .
The parts (a)-(f) follow from the respective restrictions (a)-(f) on (a n , c n ) stated in the Definition 1.
second part of Lemma 2 is a direct implication of the definition of C n .(b)-(f).The rest of the stated results follows immediately from the restrictions (b)-(f) imposed on (a n , c n ) in Definition 1.