Higher-order Galton–Watson processes with immigration having finite second moment (also called Generalized Integer-valued AutoRegressive (GINAR) processes) have been introduced by Latour [14, equation (1.1)]. Pénisson and Jacob [16] used higher-order Galton–Watson processes (without immigration) for studying the decay phase of an epidemic, and, as an application, they investigated the Bovine Spongiform Encephalopathy epidemic in Great Britain after the 1988 feed ban law. As a continuation, Pénisson [15] introduced estimators of the so-called infection parameter in the growth and decay phases of an epidemic. Recently, Kashikar and Deshmukh [12, 13] and Kashikar [11] used second order Galton–Watson processes (without immigration) for modeling the swine flu data for Pune, India and La-Gloria, Mexico. Kashikar and Deshmukh [12] also studied their basic probabilistic properties such as a formula for their probability generator function, probability of extinction, long run behavior and conditional least squares estimation of the offspring means.

Let Z+ , N , R and R+ denote the set of non-negative integers, positive integers, real numbers and non-negative real numbers, respectively. The natural basis of Rd will be denoted by {e1,…,ed} . For x∈R , the integer part of x is denoted by ⌊x⌋ . Every random variable will be defined on a probability space (Ω,A,P) . Convergence in distribution and equality in distributions of random variables or stochastic processes is denoted by ⟶D and =D , respectively.

First, we recall the Galton–Watson process with immigration, which assumes that an individual can reproduce only once during its lifetime at age 1, and then it dies immediately. The initial population size at time 0 will be denoted by X0 . For each n∈N , the population consists of the offsprings born at time n and the immigrants arriving at time n. For each n,i∈N , the number of offsprings produced at time n by the ith individual of the (n−1)th generation will be denoted by ξn,i . The number of immigrants in the nth generation will be denoted by εn . Then, for the population size Xn of the nth generation, we have (1) Xn= ∑i=1Xn−1ξn,i+εn,n∈N, where ∑i=10:=0 . Here {X0,ξn,i,εn:n,i∈N} are supposed to be independent non-negative integer-valued random variables, and {ξn,i:n,i∈N} and {εn:n∈N} are supposed to consist of identically distributed random variables, respectively. If εn=0 , n∈N , then we say that (Xn)n∈Z+ is a Galton–Watson process (without immigration).

Next, we introduce the second-order Galton–Watson branching model with immigration. In this model we suppose that an individual reproduces at age 1 and also at age 2, and then it dies immediately. For each n∈N , the population consists again of the offsprings born at time n and the immigrants arriving at time n. For each n,i,j∈N , the number of offsprings produced at time n by the ith individual of the (n−1)th generation and by the jth individual of the (n−2)th generation will be denoted by ξn,i and ηn,j , respectively, and εn denotes the number of immigrants in the nth generation. Then, for the population size Xn of the nth generation, we have (2) Xn= ∑i=1Xn−1ξn,i+ ∑j=1Xn−2ηn,j+εn,n∈N, where X−1 and X0 are non-negative integer-valued random variables (the initial population sizes). Here {X−1,X0,ξn,i,ηn,j,εn:n,i,j∈N} are supposed to be non-negative integer-valued random variables such that {(X−1,X0),ξn,i,ηn,j,εn:n,i,j∈N} are independent, and {ξn,i:n,i∈N} , {ηn,j:n,j∈N} and {εn:n∈N} are supposed to consist of identically distributed random variables, respectively. Note that the number of individuals alive at time n∈Z+ is Xn+Xn−1 , which can be larger than the population size Xn of the nth generation, since the individuals of the population at time n−1 are still alive at time n, because they can reproduce also at age 2. The stochastic process (Xn)n⩾−1 given by (2) is called a second-order Galton–Watson process with immigration or a Generalized Integer-valued AutoRegressive process of order 2 (GINAR(2) process), see, e.g., Latour [14]. Especially, if ξ1,1 and η1,1 are Bernoulli distributed random variables, then (Xn)n⩾−1 is also called an Integer-valued AutoRegressive process of order 2 (INAR(2) process), see, e.g., Du and Li [8]. If ε1=0 , then we say that (Xn)n⩾−1 is a second-order Galton–Watson process without immigration, introduced and studied by Kashikar and Deshmukh [12] as well.

The process given in (2) with the special choice η1,1=0 gives back the process given in (1), which will be called a first-order Galton–Watson process with immigration to make a distinction.

For notational convenience, let ξ, η and ε be random variables such that ξ=Dξ1,1 , η=Dη1,1 and ε=Dε1 , and put mξ:=E(ξ)∈[0,∞] , mη:=E(η)∈[0,∞] and mε:=E(ε)∈[0,∞] .

If (Xn)n∈Z+ is a (first-order) Galton–Watson process with immigration such that mξ∈(0,1) , P(ε=0)<1 and ∑j=1∞P(ε=j)log(j)<∞ , then the Markov process (Xn)n∈Z+ admits a unique stationary distribution (see, e.g., Quine [17]), i.e., the distribution of the initial value X0 can be uniquely chosen so that the process becomes strongly stationary. If ε is regularly varying with index α∈(0,∞) , i.e., P(ε>x)∈(0,∞) x)\in (0,\infty )$]]> for all x∈(0,∞) , and limx→∞P(ε>qx)P(ε>x)=q−αfor allq∈(0,∞), qx)}{\mathbb{P}(\varepsilon >x)}={q^{-\alpha }}\hspace{2em}\text{for all}\hspace{5pt}q\in (0,\infty )\text{,}\]]]> then, by Lemma A.3, ∑j=1∞P(ε=j)log(j)<∞ . The content of Theorem 2.1.1 in Basrak et al. [4] is the following statement.

Note that in case of α=1 and mε=∞ Basrak et al. [4, Theorem 2.1.1] assume additionally that ε is consistently varying (or in other words intermediate varying), but, eventually, this follows from the fact that ε is regularly varying. Basrak et al. [4, Remark 2.2.2] derived the result of Theorem 1 also for α∈[2,3) under the additional assumption E(ξ3)<∞ (not mentioned in the paper), and they remarked that the same applies to all α∈[3,∞) (possibly under an additional moment assumption E(ξ⌊α⌋+1)<∞ ).

In Barczy et al. [3] we study regularly varying non-stationary (first-order) Galton–Watson processes with immigration.

If (Xn)n⩾−1 is a second-order Galton–Watson process with immigration such that mξ,mη∈(0,1) with mξ+mη<1 , P(ε=0)<1 and ∑j=1∞P(ε=j)log(j)<∞ , then the distribution of the initial values (X0,X−1) can be uniquely chosen so that the process becomes strongly stationary, see Lemma 3.

The main result of the paper is the following analogue of Theorem 1.

Note that λ+ and λ− are the eigenvalues of the offspring mean matrix given in (8) of a corresponding 2-type Galton–Watson process with immigration. Note that for all k∈Z+ , we have mk=E(Vk,0) , where (Vn,0)n⩾−1 is a second-order Galton–Watson process (without immigration) with the initial values V0,0=1 and V−1,0=0 , and with the same offspring distributions as (Xn)n⩾−1 , see (9). Consequently, the series ∑i=0∞miα appearing in Theorem 2 is convergent, since for each i∈N , we have mi=E(Vi,0)⩽λ+i<1 by (10) and the assumption mξ+mη<1 .

Our technique and result might be extended to p-th order Galton–Watson branching processes with immigration. More generally, one can pose an open problem, namely, under what conditions on the offspring and immigration distributions of a general p-type Galton–Watson branching process with immigration, its unique (p-dimensional) stationary distribution is jointly regularly varying. We also note that there is a vast literature on tail behavior of regularly varying time series (see, e.g., Hult and Samorodnitsky [10]), however, the available results do not seem to be applicable for describing the tail behavior of the stationary distribution for regularly varying branching processes with immigration. The link between GINAR and autoregressive processes is that their autocovariance functions are identical under finite second moment assumptions, but we cannot see that it would imply anything for the tail behavior of a GINAR process knowing the tail behaviour of a corresponding autoregressive process. Further, in our situation the second moment is infinite, so the autocovariance function is not defined.

Very recently, Bősze and Pap [5] have studied regularly varying non-stationary second-order Galton–Watson processes with immigration. They have found some sufficient conditions on the initial, the offspring and the immigration distributions of a non-stationary second-order Galton–Watson process with immigration under which the distribution of the process in question is regularly varying at any fixed time. The results in Bősze and Pap [5] can be considered as extensions of the results in Barczy et al. [3] on not necessarily stationary (first-order) Galton–Watson processes with immigration. Concerning the results in Bősze and Pap [5] and in the present paper, there is no overlap, for more details see Remark 1.

The paper is organized as follows. In Section 2 we present preliminaries. First we recall a representation of a second-order Galton–Watson process without or with immigration as a (special) 2-type Galton–Watson process without or with immigration, respectively. Then, we derive an explicit formula for the expectation of a second-order Galton–Watson process with immigration at time n and describe its asymptotic behavior as n→∞ , and, assuming finiteness of the second moments of the offspring distributions, we give an estimation of the second moment of a second-order Galton–Watson process (without immigration). Next, we recall sufficient conditions for the existence of a unique stationary distribution for a 2-type Galton–Watson process with immigration, and a representation of this stationary distribution. Applying these results to the special 2-type Galton–Watson process with immigration belonging to the class of second-order Galton–Watson processes with immigration, we obtain sufficient conditions for the existence of a unique distribution of the initial values (X0,X−1) such that the process becomes strongly stationary, see Lemma 3. Section 3 is devoted to the proof of Theorem 2. In the course of the proof sufficient conditions are given under which the distribution of a second-order Galton–Watson processes (without immigration) (Xn)n⩾−1 at a fixed time is regularly varying provided that X0 is regularly varying and X−1=0 , see Proposition 1. In the Appendix we collect some results on regularly varying functions and distributions, to name a few of them: convolution property, Karamata’s theorem and Potter’s bounds. Note that the ArXiv version [2] of this paper contains more details, proofs and appendices.

First, we recall a representation of a second-order Galton–Watson process without or with immigration as a (special) 2-type Galton–Watson process without or with immigration, respectively. Let (Xn)n⩾−1 be a second-order Galton–Watson process with immigration given in (2), and let us introduce the random vectors (4) Yn:=Yn,1Yn,2:=XnXn−1,n∈Z+. Then we have (5) Yn= ∑i=1Yn−1,1ξn,i1+ ∑j=1Yn−1,2ηn,j0+εn0,n∈N, hence (Yn)n∈Z+ is a (special) 2-type Galton–Watson process with immigration and with initial vector Y0=X0X−1. In fact, the type 1 and 2 individuals are identified with individuals of age 0 and 1, respectively, and for each n,i,j∈N , at time n, the ith individual of type 1 of the (n−1)th generation produces ξn,i individuals of type 1 and exactly one individual of type 2, and the jth individual of type 2 of the (n−1)th generation produces ηn,j individuals of type 1 and no individual of type 2.

The representation (5) works backwards as well, namely, let (Yk)k∈Z+ be a special 2-type Galton–Watson process with immigration given by (6) Yk= ∑j=1Yk−1,1ξk,j,1,11+ ∑j=1Yk−1,2ξk,j,2,10+εk,10,k∈N, where Y0 is a 2-dimensional integer-valued random vector. Here, for each k,j∈N and i∈{1,2} , ξk,j,i,1 denotes the number of type 1 offsprings in the kth generation produced by the jth offspring of the (k−1)th generation of type i, and εk denotes the number of type 1 immigrants in the kth generation. For the second coordinate process of (Yk)k∈Z+ , we get Yk,2=Yk−1,1 , k∈N , and substituting this into (6), the first coordinate process of (Yk)k∈Z+ satisfies Yk,1= ∑j=1Yk−1,1ξk,j,1,1+ ∑j=1Yk−2,1ξk,j,2,1+εk,1,k⩾2. Thus, the first coordinate process of (Yk)k∈Z+ given by (6) satisfies equation (2) with Xn:=Yn,1 , ξn,i:=ξn,i,1,1 , ηn,j:=ξn,j,2,1 , εn:=εn,1 , n,i,j∈N , and with the initial values X0:=Y0,1 and X−1:=Y0,2 , i.e., it is a second-order Galton–Watson process with immigration.

Note that, for a second-order Galton–Watson process (Xn)n⩾−1 (without immigration), the additive (or branching) property of a 2-type Galton–Watson process (without immigration) (see, e.g. in Athreya and Ney [1, Chapter V, Section 1]), together with the law of total probability, for each n∈N , implies (7) Xn=D ∑i=1X0ζi,0(n)+ ∑j=1X−1ζj,−1(n), where {(X0,X−1),ζi,0(n),ζj,−1(n):i,j∈N} are independent random variables such that {ζi,0(n):i∈N} are independent copies of Vn,0 and {ζj,−1(n):j∈N} are independent copies of Vn,−1 , where (Vk,0)k⩾−1 and (Vk,−1)k⩾−1 are second-order Galton–Watson processes (without immigration) with initial values V0,0=1 , V−1,0=0 , V0,−1=0 and V−1,−1=1 , and with the same offspring distributions as (Xk)k⩾−1 .

Moreover, if (Xn)n⩾−1 is a second-order Galton–Watson process with immigration, then for each n∈N , we have Xn=V0(n)(X0,X−1)+ ∑i=1nVi(n−i)(εi,0), where {V0(n)(X0,X−1),Vi(n−i)(εi,0):i∈{1,…,n}} are independent random variables such that V0(n)(X0,X−1) represents the number of newborns at time n, resulting from the initial individuals X0 at time 0 and X−1 at time −1 , and for each i∈{1,…,n} , Vi(n−i)(εi,0) represents the number of newborns at time n, resulting from the immigration εi at time i, see the ArXiv version [2] of this paper.

Our next aim is to derive an explicit formula for the expectation of a subcritical second-order Galton–Watson process with immigration at time n and to describe its asymptotic behavior as n→∞ .

Recall that ξ, η and ε are random variables such that ξ=Dξ1,1 , η=Dη1,1 and ε=Dε1 , and we put mξ=E(ξ)∈[0,∞] , mη=E(η)∈[0,∞] and mε=E(ε)∈[0,∞] . If mξ∈R+ , mη∈R+ , mε∈R+ , E(X0)∈R+ and E(X−1)∈R+ , then (2) implies E(Xn|Fn−1X)=Xn−1mξ+Xn−2mη+mε,n∈N, where FnX:=σ(X−1,X0,…,Xn) , n∈Z+ . Consequently, E(Xn)=mξE(Xn−1)+mηE(Xn−2)+mε,n∈N, which can be written in the matrix form E(Xn)E(Xn−1)=Mξ,ηE(Xn−1)E(Xn−2)+mε0,n∈N, with (8) Mξ,η:=mξmη10. Note that Mξ,η is the mean matrix of the 2-type Galton–Watson process (Yn)n∈Z+ given in (4). Thus, we conclude E(Xn)E(Xn−1)=Mξ,ηnE(X0)E(X−1)+ ∑k=1nMξ,ηn−kmε0,n∈N. Hence, the asymptotic behavior of the sequence (E(Xn))n∈N depends on the asymptotic behavior of the powers (Mξ,ηn)n∈N , which is related to the spectral radius ϱ of Mξ,η . The matrix Mξ,η has eigenvalues λ+ and λ− given in (3) and satisfying λ+∈R+ and λ−∈[−λ+,0] , hence the spectral radius of Mξ,η is ϱ=λ+ . If (Xn)n⩾−1 is a second-order Galton–Watson process with immigration such that mξ∈R+ and mη∈R+ , then (Xn)n⩾−1 is called subcritical, critical or supercritical if ϱ<1 , ϱ=1 or ϱ>1 1$]]>, respectively. It is easy to check that a second-order Galton–Watson process with immigration is subcritical, critical or supercritical if and only if mξ+mη<1 , mξ+mη=1 or mξ+mη>1 1$]]>, respectively.

The first moment of a subcritical second-order Galton–Watson process (Xn)n⩾−1 (without immigration) can be estimated by (10). Next, we present an auxiliary lemma on an estimation of the second moment of a subcritical second-order Galton–Watson process (without immigration).

The proofs of Lemmata 1 and 2 together with statements in the critical and supercritical cases can be found in the ArXiv version [2] of this paper.

Next, we recall 2-type Galton–Watson processes with immigration. For each k,j∈Z+ and i,ℓ∈{1,2} , the number of individuals of type i born or arrived as immigrants in the kth generation will be denoted by Xk,i , the number of type ℓ offsprings produced by the jth individual who is of type i belonging to the (k−1)th generation will be denoted by ξk,j,i,ℓ , and the number of type i immigrants in the kth generation will be denoted by εk,i . Then we have Xk,1Xk,2= ∑j=1Xk−1,1ξk,j,1,1ξk,j,1,2+ ∑j=1Xk−1,2ξk,j,2,1ξk,j,2,2+εk,1εk,2,k∈N. Here {X0,ξk,j,i,εk:k,j∈N,i∈{1,2}} are supposed to be independent, and {ξk,j,1:k,j∈N} , {ξk,j,2:k,j∈N} and {εk:k∈N} are supposed to consist of identically distributed random vectors, where X0:=X0,1X0,2,ξk,j,i:=ξk,j,i,1ξk,j,i,2,εk:=εk,1εk,2. For notational convenience, let ξ1 , ξ2 and ε be random vectors such that ξ1=Dξ1,1,1 , ξ2=Dξ1,1,2 and ε=Dε1 , and put mξ1:=E(ξ1)∈[0,∞]2 , mξ2:=E(ξ2)∈[0,∞]2 , mε:=E(ε)∈[0,∞]2 , and Mξ:=mξ1mξ2∈[0,∞]2×2. We call Mξ the offspring mean matrix, and note that many authors define the offspring mean matrix as Mξ⊤ . If mξ1∈R+2 , mξ2∈R+2 , the spectral radius of Mξ is less than 1, Mξ is primitive, i.e., there exists m∈N such that Mξm∈R++2×2 , P(ε=0)<1 and E(1{ε≠0}log((e1+e2)⊤ε))<∞ , then, by the Theorem in Quine [17], there exists a unique stationary distribution π for (Xn)n∈Z+ . As a consequence of formula (16) for the probability generating function of π in Quine [17], we have ∑i=0nVi(i)(εi)⟶Dπasn→∞, where (Vk(i)(εi))k∈Z+ , i∈Z+ , are independent copies of a 2-type Galton–Watson process (Vk(ε))k∈Z+ (without immigration) with an initial vector V0(ε)=ε and with the same offspring distributions as (Xk)k∈Z+ . Consequently, we have (11) ∑i=0∞Vi(i)(εi)=Dπ, where the series ∑i=0∞Vi(i)(εi) converges with probability 1, see, e.g., Heyer [9, Theorem 3.1.6]. The above representation of the stationary distribution π for (Xn)n∈Z+ can be interpreted in a way that we consider independent 2-type Galton–Watson processes without immigration such that the ith one admits initial vector εi , i∈Z+ , evaluate the ith 2-type Galton–Watson processes at time point i, and then sum up all these random variables.

Next, we give sufficient conditions for the strong stationarity of a subcritical second-order Galton–Watson process with immigration.

Note also that (Xn)n⩾−1 is only a second-order Markov chain, but not a Markov chain. Remark 1.