On aggregation of multitype Galton-Watson branching processes with immigration

Limit behaviour of temporal and contemporaneous aggregations of independent copies of a stationary multitype Galton-Watson branching process with immigration is studied in the so-called iterated and simultaneous cases, respectively. In both cases, the limit process is a zero mean Brownian motion with the same covariance function under third order moment conditions on the branching and immigration distributions. We specialize our results for generalized integer-valued autoregressive processes and single-type Galton-Watson processes with immigration as well.


Introduction
The field of temporal and contemporaneous aggregations of independent stationary stochastic processes is an important and very active research area in the empirical and theoretical statistics and in other areas as well. The scheme of contemporaneous (also called cross-sectional) aggregation of random-coefficient autoregressive processes of order 1 was firstly proposed by Robinson [16] and Granger [4] in order to obtain the long memory phenomena in aggregated time series. For surveys on papers dealing with the aggregation of different kinds of stochastic processes, see, e.g., Pilipauskaitė and Surgailis [13], Jirak [8, page 512] or the arXiv version of Barczy et al. [2].
In this paper we study the limit behaviour of temporal (time) and contemporaneous (space) aggregations of independent copies of a strictly stationary multitype Galton-Watson branching process with immigration in the so-called iterated and simultaneous cases, respectively. According to our knowledge, the aggregation of general multitype Galton-Watson branching processes with immigration has not been considered in the literature so far. To motivate the fact that the aggregation of branching processes could be an important topic, now we present an interesting and relevant example, where the phenomena of aggregation of this kind of processes may come into play. A usual Integer-valued AutoRegressive (INAR) process of order 1, (X k ) k 0 , can be used to model migration, which is quite a big issue nowadays all over the world. More precisely, given a camp, for all k 0, the random variable X k can be interpreted as the number of migrants to be present in the camp at time k, and every migrant will stay in the camp with probability α ∈ (0, 1) indepedently of each other (i.e., with probability 1 − α each migrant leaves the camp) and at any time k 1 new migrants may come to the camp. Given several camps in a country, we may suppose that the corresponding INAR processes of order 1 share the same parameter α and they are independent. So, the temporal and contemporaneous aggregations of these INAR processes of order 1 is the total usage of the camps in terms of the number of migrants in the given country in a given time period, and this quantity may be worth studying.
The present paper is organized as follows. In Section 2 we formulate our main results, namely the iterated and simultaneous limit behaviour of time-and space-aggregated independent stationary p-type Galton-Watson branching processes with immigration is described (where p 1), see Theorems 2.6 and 2.7. The limit distributions in these limit theorems coincide, namely, it is a p-dimensional zero mean Brownian motion with a covariance function depending on the expectations and covariances of the offspring and immigration distributions. In the course of the proofs of our results, in Lemma 2.3, we prove that for a subcritical, positively regular multitype Galton-Watson branching process with nontrivial immigration, its unique stationary distribution admits finite α th moments provided that the branching and immigration distributions have finite α th moments, where α ∈ {1, 2, 3}. In case of α ∈ {1, 2}, Quine [14] contains this result, however in case of α = 3, we have not found any precise proof in the literature for it, it is something like a folklore, so we decided to write down a detailed proof. As a by-product, we obtain an explicit formula for the third moment in question. Section 3 is devoted to the special case of generalized INAR processes, especially to single-type Galton-Watson branching processes with immigration. All of the proofs can be found in Section 4.

Aggregation of multitype Galton-Watson branching processes with immigration
Let Z + , N, R, R + , and C denote the set of non-negative integers, positive integers, real numbers, non-negative real numbers, and complex numbers, respectively. For all d ∈ N, the d × d identity matrix is denoted by I d . The standard basis in R d is denoted by {e 1 , . . . , e d }.
For v ∈ R d , the Euclidean norm is denoted by v , and for A ∈ R d×d , the induced matrix norm is denoted by A as well (with a little abuse of notation). All the random variables will be defined on a probability space (Ω, F , P).
Let (X k = [X k,1 , . . . , X k,p ] ⊤ ) k∈Z + be a p-type Galton-Watson branching process with immigration. For each k, ℓ ∈ Z + and i, j ∈ {1, . . . , p}, the number of j-type individuals in the k th generation will be denoted by X k,j , the number of j-type offsprings produced by the ℓ th individual belonging to type i of the (k − 1) th generation will be denoted by ξ (i,j) k,ℓ , and the number of immigrants of type i in the k th generation will be denoted by ε (i) k . Then we have . . , p} are supposed to be independent Z p + -valued random vectors. Note that we do not assume independence among the components of these vectors. Moreover, for all i ∈ {1, . . . , p}, k,ℓ : k, ℓ ∈ N} and {ε, ε k : k ∈ N} are supposed to consist of identically distributed random vectors, respectively.

Let us introduce the notations m
for i, j ∈ {1, . . . , p}, provided that the expectations and covariances in question are finite. Let ̺(M ξ ) denote the spectral radius of M ξ , i.e., the maximum of the modulus of the eigenvalues of M ξ . The process (X k ) k∈Z + is called subcritical, critical or supercritical if ̺(M ξ ) is smaller than 1, equal to 1 or larger than 1, respectively. The matrix M ξ is called primitive if there is a positive integer n ∈ N such that all the entries of M n ξ are positive. The process (X k ) k∈Z + is called positively regular if M ξ is primitive. In what follows, we suppose that For further application, we define the matrix provided that the covariances in question are finite.
2.1 Remark. Note that the matrix (I p − M ξ ) −1 , which appears in (2.3) and throughout the paper, exists. Indeed, λ ∈ C is an eigenvalue of I p − M ξ if and only if 1 − λ is that of M ξ . Therefore, since ρ(M ξ ) < 1, all eigenvalues of I p − M ξ are non-zero. This means that det(I p − M ξ ) = 0, so (I p − M ξ ) −1 does exist. One could also refer to Corollary 5.6.16 and Lemma 5.6.10 in Horn and Johnson [6]. ✷ 2.2 Remark. Note that V is symmetric and positive semidefinite, since v (i,j) = v (j,i) , i, j ∈ {1, . . . , p}, and for all x ∈ R p , Here due to the fact that (I p − M ξ ) −1 m ε is nothing else but the expectation vector of the unique stationary distribution of (X k ) k∈Z + , see the discussion below and formula (4.4). ✷ Under (2.2), by the Theorem in Quine [14], there is a unique stationary distribution π for (X k ) k∈Z + . Indeed, under (2.2), M ξ is irreducible following from the primitivity of M ξ , see Definition 8.5.0 and Theorem 8.5.2 in Horn and Johnson [6]. For the definition of irreducibility, see Horn Section 3]. For the definition of aperiodicity (also called acyclicity), see, e.g., the Introduction of Danka and Pap [3]. Finally, since m ε ∈ R p + \ {0}, the probability generator function of ε at 0 is less than 1, and so one can apply the Theorem in Quine [14].
For each α ∈ N, we say that the α th moment of a random vector is finite if all of its mixed moments of order α are finite.
In what follows, we suppose (2.2) and that the distribution of X 0 is the unique stationary distribution π, hence the Markov chain (X k ) k∈Z + is strictly stationary. Recall that, by (2.1) in Quine and Durham [15], for any measurable function f : First we consider a simple aggregation procedure. For each N ∈ N, consider the stochastic process S (N ) = (S (N ) k ) k∈Z + given by is a sequence of independent copies of the strictly stationary p-type Galton-Watson process (X k ) k∈Z + with immigration. Here we point out that we consider so-called idiosyncratic immigrations, i.e., the immigrations belonging to X (j) , j ∈ N, are independent.
We will use where X = (X k ) k∈Z + is a stationary p-dimensional zero mean Gaussian process with covariances We note that using formula (4.6) presented later on, one could give an explicit formula for Var(X 0 ) (not containing an infinite series).

Proposition.
If all entries of the vectors ξ (i) , i ∈ {1, . . . , p}, and ε have finite third moments, then 2.6 Theorem. If all entries of the vectors ξ (i) , i ∈ {1, . . . , p}, and ε have finite second moments, then where B = (B t ) t∈R + is a p-dimensional zero mean Brownian motion satisfying Var(B 1 ) = V . If all entries of the vectors ξ (i) , i ∈ {1, . . . , p}, and ε have finite third moments, then Theorem. If all entries of the vectors ξ (i) , i ∈ {1, . . . , p}, and ε have finite third moments, then if both n and N converge to infinity (at any rate), where B = (B t ) t∈R + is a p-dimensional zero mean Brownian motion satisfying Var(B 1 ) = V .
A key ingredient of the proofs is the fact that (X k −E(X k )) k∈Z + can be rewritten as a stable first order vector autoregressive process with coefficient matrix M ξ and with heteroscedastic innovations, see (4.14).

A special case: aggregation of GINAR processes
We devote this section to the analysis of aggregation of Generalized Integer-Valued Autoregressive processes of order p ∈ N (GINAR(p) processes), which are special cases of p-type Galton-Watson branching processes with immigration introduced in (2.1). For historical fidelity, we note that it was Latour [11] who introduced GINAR(p) processes as generalizations of INAR(p) processes. This class of processes became popular in modelling integer-valued time series data such as the daily number of claims at an insurance company. In fact, a GINAR(1) process is a (general) single type Galton-Watson branching processes with immigration.
A GINAR(p) process can be embedded in a p-type Galton-Watson branching process with In what follows, we reformulate the classification of GINAR(p) processes in terms of the expectations of the offspring distributions.
3.1 Remark. In case of a GINAR(p) process, one can show that ϕ, the characteristic polynomial of the matrix M ξ , has the form Recall that ̺(M ξ ) denotes the spectral radius of M ξ , i.e., the maximum of the modulus of the eigenvalues of M ξ . If E(ξ (p,1) ) > 0, then, by the proof of Proposition 2.2 in Barczy et al. [1], the characteristic polynomial ϕ has just one positive root, ✷ Next, we specialize the matrix V , defined in (2.3), in case of a subcritical GINAR(p) process.
3.2 Remark. In case of a GINAR(p) process, the vectors for i, j ∈ {1, . . . , p} are all zero vectors except for the case i = j = 1. Therefore, in case of .
✷ Finally, we specialize the limit distribution in Theorems 2.6 and 2.7 in case of a subcritical GINAR(p) process.
3.3 Remark. Let us note that in case of p = 1 and E(ξ (1,1) ) < 1 (yielding that the corresponding GINAR(1) process is subcritical), the limit process in Theorems 2.6 and 2.7 can be written as

Proofs
Proof of Lemma 2.3. Let (Z k ) k∈Z + be a p-type Galton-Watson branching process without immigration, with the same offspring distribution as (X k ) k∈Z + , and with Z 0 D = ε. Then the stationary distribution π of (X k ) k∈Z + admits the representation where (Z (n) k ) k∈Z + , n ∈ Z + , are independent copies of (Z k ) k∈Z + . This is a consequence of formula (16) for the probability generating function of π in Quine [14]. It is convenient to calculate moments of Kronecker powers of random vectors. We will use the notation A ⊗ B for the Kronecker product of the matrices A and B, and we put A ⊗2 := A ⊗ A and For each n ∈ Z + , we have where (Y k ) k∈Z + is a Galton-Watson branching process with the same offspring and immigration distributions as (X k ) k∈Z + , and with Y 0 = 0. This can be checked comparing their probability generating functions taking into account formula (3) in Quine [14] as well. Consequently, we conclude For each n ∈ N, using (2.1), we obtain Taking the expectation, we get , see Corollary 5.6.16 and Lemma 5.6.10 in Horn and Johnson [6]. Consequently, by (4.1), the first moment of π is finite, and Now we suppose that the second moments of ξ (i) , i ∈ {1, . . . , p}, and ε are finite. For each n ∈ N, using again (2.1), we obtain Indeed, using the mixed-product property (A ⊗ B)(C ⊗ D) = (AC) ⊗ (BD) for matrices of such size that one can form the matrix products AC and BD, we have Consequently, we obtain Taking the expectation, we get Using also (4.3), we obtain Taking into account Y 0 = 0, we obtain Taking into account ̺(M ξ ) < 1, we conclude ̺(A 2 ) = ̺(M ξ ) < 1, and, by (4.1), the second moment of π is finite, and Now we suppose that the third moments of ξ (i) , i ∈ {1, . . . , p}, and ε are finite. For each n ∈ N, using again (2.1), we obtain We have which can be written in the form Moreover, where E(ξ (i) ⊗ ε ⊗ ξ (i) ) is finite, since there exists a permutation matrix P ∈ R p 2 ×p 2 such that u ⊗ v = P (v ⊗ u) for all u, v ∈ R p (see, e.g., Henderson and Searle [5, formula (6)]), hence Hence (4.8) S n,2 = A 3,2 Y ⊗2 n−1 + A Further, Consequently, we have 3,2 and A 3,1 := A 3,1 + A 3,1 + A 3,1 . Taking the expectation, we get Summarizing, we obtain Taking into account Y 0 = 0, we obtain We have ̺( and, by (4.1), the third moment of π is finite, and (4.10) where F X k := σ(X 0 , . . . , X k ), k ∈ Z + . Consequently, (4.11) and, by (4.4), Then E(U k | F X k−1 ) = 0, k ∈ N, and using the independence of ξ (i) k,ℓ , ε k : k, ℓ ∈ N, i ∈ {1, . . . , p} , we have . . , p} and k ∈ N, where [U k,1 , . . . , U k,p ] ⊤ := U k , k ∈ N. For each k ∈ N, using X k = M ξ X k−1 + m ε + U k and (4.11), we obtain (4.14) Consequently, Taking the expectation, by (4.12) and (4.13), we conclude Under the conditions of the proposition, by Lemma 2.3, the unique stationary distribution π has a finite second moment, hence, using again the stationarity of (X k ) k∈Z + , for each N ∈ N, we get  (2.6). Moreover, by (4.14), Taking the expectation, we conclude Hence, by induction, we obtain the formula for Cov(X 0 , X k ). The statement will follow from the multidimensional central limit theorem. Due to the continuous mapping theorem, it is sufficient to show the convergence N −1/2 (S k )) ⊤ ⊤ , j ∈ N, are independent, identically distributed having zero mean vector and covariances Cov(X (j) for j ∈ N, ℓ 1 , ℓ 2 ∈ {0, 1, . . . , k}, ℓ 1 ℓ 2 , following from the strict stationarity of X (j) and from (2.5). ✷

Proof of Proposition 2.5. It is known that
are martingale differences with respect to the filtration (F X k ) k∈Z + . The functional martingale central limit theorem can be applied, see, e.g., Jacod and Shiryaev [7, Theorem VIII.3.33]. Indeed, using (4.13) and the fact that the first moment of X 0 exists and is finite, by (2.4), for each t ∈ R + , and i, j ∈ {1, . . . , p}, we have and hence the convergence holds in probability as well. Moreover, the conditional Lindeberg condition holds, namely, for all δ > 0, where the last inequality follows by Proposition 3.3 of Nedényi [12], and the almost sure convergence is a consequence of (2.4), since, under the third order moment assumptions in Proposition 2.5, by Lemma 2.
U j converges in L 1 and hence in probability to 0 as n → ∞, since by (4.13), and hence V j,j → 0 as n → ∞, (4.20) since, applying (4.16) for ⌊nt⌋ k 0 , we have Consequently, by Slutsky's lemma, where B = (B t ) t∈R + is a p-dimensional zero mean Brownian motion satisfying Var(B 1 ) = V , as desired. ✷ Proof of Theorem 2.6. First, we prove (2.8). For all N, m ∈ N and all t 1 , . . . , t m ∈ R + , by Proposition 2.5 and the continuity theorem, we have Now, we turn to prove (2.7). For all n ∈ N and for all t 1 , . . . , t m ∈ R + with t 1 < . . . < t m , m ∈ N, by Proposition 2.4 and by the continuous mapping theorem, we have where (X k ) k∈Z + is the p-dimensional zero mean stationary Gaussian process given in Proposition 2.4 and, by (2.5), Var , where q 2 ℓ=q 1 := 0 for all q 2 < q 1 , q 1 , q 2 ∈ N. By the continuity theorem, for all θ 1 , . . . , θ m ∈ R p , m ∈ N, we conclude Indeed, for all s, t ∈ R + with s < t, we have (4.16). It remains to show that hence, by (4.22), the right-hand side of the equation (4.21) can be written as , and we conclude (4.21). This implies the convergence (2.7). ✷ Proof of Theorem 2.7. As n and N converge to infinity simultaneously, (2.9) is equivalent to (nN n ) − 1 2 S (Nn,n) D −→ (I p − M ξ ) −1 B as n → ∞ for any sequence (N n ) n∈N of positive integers such that lim n→∞ N n = ∞. As we have seen in the proof of Proposition 2.5, for each j ∈ N, U (j) are martingale differences with respect to the filtration (F X (j) k ) k∈Z + . We are going to apply the functional martingale central limit theorem, see, e.g., Jacod and Shiryaev [7, Theorem VIII. ), and for each k = ℓN n +r with ℓ ∈ Z + and r ∈ {1, . . . , N n }, we have F ), and for k = ℓN n + r with ℓ ∈ Z + and r ∈ {1, . . . , N n }, we have Y ℓ+1 . Next we check that for each n ∈ N, (V (n) k ) k∈N is a sequence of martingale differences with respect to (F (n) k ) k∈Z + . We will use that E(ξ | σ(G 1 ∪ G 2 )) = E(ξ | G 1 ) for a random vector ξ and for σ-algebras G 1 ⊂ F and G 2 ⊂ F such that σ(σ(ξ) ∪ G 1 ) and G 2 are independent and E( ξ ) < ∞. For each k = ℓN n + 1 with ℓ ∈ Z + , we have In a similar way, for each k = ℓN n + r with ℓ ∈ Z + and r ∈ {2, . . . , N n }, we have We want to obtain a functional central limit theorem for the sequence First, we calculate the conditional variance matrix of V (n) k . If k = ℓN n + 1 with ℓ ∈ Z + , then In a similar way, if k = ℓN n + r with ℓ ∈ Z + and r ∈ {2, . . . , N n }, then Consequently, for each n ∈ N and t ∈ R + , we have as n → ∞. Indeed, the equality follows by (4.13), and for the convergence in probability, note that lim n→∞ ⌊nt⌋ n = t, t ∈ R + , and, by Cauchy-Schwarz inequality, E(|(X ℓ 1 −1,m 1 − E(X 0,m 1 ))(X ℓ 2 −1,m 2 − E(X 0,m 2 ))|) Var(X ℓ 1 −1,m 1 ) Var(X ℓ 2 −1,m 2 ) Var(X 0,m 1 ) Var(X 0,m 2 ) → 0 as n → ∞, where we used that Q p i=1 p j=1 |q i,j | for every matrix Q = (q i,j ) p i,j=1 ∈ R p×p . Moreover, in a similar way, the conditional Lindeberg condition holds, namely, for all δ > 0,  20), where for the last inequality we used (4.19). This completes the proof. ✷