Asymptotic behavior of CLS estimators for 2-type doubly symmetric critical Galton Watson processes with immigration

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1 Bernoulli (4), 4, DOI:.35/3-BEJ556 Asymptotic behavior of CLS estimators for -type doubly symmetric critical Galton Watson processes with immigration MÁRTON ISÁNY, KRISTÓF KÖRMENDI,* and GYULA A,** University of Debrecen, Faculty of Informatics, Department of Information Technology, f., H-4 Debrecen, Hungary. University of Szeged, Faculty of Science, Bolyai Institute, Department of Stochastics, Aradi vértanú tere, H-67 Szeged, Hungary. * ormendi@math.u-szeged.hu; ** papgy@math.u-szeged.hu In this paper, the asymptotic behavior of the conditional least squares (CLS) estimators of the offspring means (α, β) and of the criticality parameter ϱ := α + β for a -type critical doubly symmetric positively regular Galton Watson branching process with immigration is described. Keywords: conditional least squares estimator; Galton Watson branching process with immigration. Introduction Asymptotic behavior of CLS estimators for critical Galton Watson processes is available only for single-type processes, see Wei and Winnici,] and Winnici ], see also the monograph of Guttorp 4]. In the present paper, the asymptotic behavior of the CLS estimators of the offspring means and criticality parameter for -type critical doubly symmetric positively regular Galton Watson process with immigration is described, see Theorem 3.. This study can be considered as the first step of examining the asymptotic behavior of the CLS estimators of parameters of multitype critical branching processes with immigration. Shete and Sriram 8] obtained convergence results for weighted CLS estimators in the ercritical case. Let us recall the results for a single-type Galton Watson branching process (X ) Z+ with immigration and with initial value X =. Suppose that it is critical, that is, the offspring mean equals. Wei and Winnici ] proved a functional limit theorem X (n) D X as n, where X t (n) := n X nt for t R +, n N, where x denotes the (lower) integer part of x R, and (X t ) t R+ is a (nonnegative) diffusion process with initial value X = and with generator Lf (x) = m ε f (x) + V xf (x), f C c (R +), where m ε denotes the immigration mean, V denotes the offspring variance, and Cc (R +) denotes the space of infinitely differentiable functions on R + with compact port. The process (X t ) t R+ can also be characterized as the unique strong solution of the stochastic differential equation (SDE) dx t = m ε dt + V X t + dw t, t R +, ISI/BS

2 48 M. Ispány, K. Körmendi and G. ap with initial value X =, where (W t ) t R+ is a standard Wiener process, and x + denotes the positive part of x R. Note that this so-called square-root process is also nown as Feller diffusion, or Cox Ingersoll Ross model in financial mathematics (see Musiela and Rutowsi 5], page 9). In fact, (4V X t ) t R+ is the square of a 4V m ε -dimensional Bessel process started at (see Revuz and Yor 7], XI..). Assuming that the immigration mean m ε is nown, for the conditional least squares estimator (CLSE) n X (X m ε ) α n (X,...,X n ) = n X of the offspring mean based on the observations X,...,X n, one can derive n ( α n (X,...,X n ) ) D X t d(x t m ε t) X t dt as n. (Wei and Winnici ] contains a similar result for the CLS estimator of the offspring mean when the immigration mean is unnown.) In Section, we recall some preliminaries on -type Galton Watson models with immigration. Section 3 contains our main results. Sections 4, 5, 6 and 7 contain the proofs. Appendix A is devoted to the CLS estimators. In Appendix B, we present estimates for the moments of the processes involved. Appendices C and D are for a version of the continuous mapping theorem and for convergence of random step processes, respectively. For a detailed discussion of the whole paper, see Ispány et al. 8].. reliminaries on -type Galton Watson models with immigration Let Z +, N, R and R + denote the set of nonnegative integers, positive integers, real numbers and non-negative real numbers, respectively. Every random variable will be defined on a fixed probability space (, A, ). For each,j Z + and i, l {, }, the number of individuals of type i in the th generation will be denoted by X,i, the number of type l offsprings produced by the jth individual who is of type i belonging to the ( )th generation will be denoted by,j,i,l, and the number of type i immigrants in the th generation will be denoted by ε,i. Then X, X, ] = X, j=,j,,,j,, ] + X, j=,j,,,j,, ] + ε, ε, Here {X,,j,i, ε :,j N,i {, }} are posed to be independent, where X := X, X, ],,j,i :=,j,i,,j,i, ], ε := ], N. (.) ε, ε, ].

3 -type critical Galton Watson processes with immigration 49 Moreover, {,j, :,j N}, {,j, :,j N} and {ε : N} are posed to consist of identically distributed random vectors. We pose E(,, )<, E(,, )< and E( ε )<. Introduce the notations m i := E(,,i ) R +, m := m m ] R +, V i := Var(,,i ) R, V := (V + V ) R, m ε := E(ε ) R +, V ε := Var(ε ) R. Note that many authors define the offspring mean matrix as m.for Z +,letf := σ(x, X,...,X ).By(.), E(X F ) = X, m + X, m + m ε = m X + m ε. (.) Consequently, E(X ) = m E(X ) + m ε, N, which implies E(X ) = m E(X ) + m j m ε, N. (.3) j= Hence, the offspring mean matrix m plays a crucial role in the asymptotic behavior of the sequence (X ) Z+. Since m has nonnegative entries, the Frobenius erron theorem (see, e.g., Horn and Johnson 7], Theorems 8.. and 8.5.) describes the behavior of the powers m as. According to this behavior, a -type Galton Watson process (X ) Z+ with immigration is referred to respectively as subcritical, critical or ercritical if ϱ<, ϱ = orϱ>, where ϱ denotes the spectral radius of the offspring mean matrix m (see, e.g., Athreya and Ney ] orquine6]). We will consider doubly symmetric -type Galton Watson processes with immigration, when the offspring mean matrix has the form ] α β m :=. (.4) β α Its spectral radius is ϱ = α + β, which will be called criticality parameter. We will focus only on positively regular doubly symmetric -type Galton Watson processes with immigration, that is, when there is a positive integer N such that the entries of m are positive (see Kesten and Stigum 3]), which is equivalent with α> and β>. For the sae of simplicity, we consider a zero start Galton Watson process with immigration, that is, we pose X =. In the sequel, we always assume m ε, otherwise X = for all N. 3. Main results In order to find CLS estimators of the criticality parameter ϱ = α + β, we introduce a further parameter δ := α β. Then α = (ϱ + δ)/ and β = (ϱ δ)/, thus the recursion (4.) can be

4 5 M. Ispány, K. Körmendi and G. ap written in the form X = ] ϱ + δ ϱ δ X ϱ δ ϱ+ δ + M + m ε, N. For each n N, a CLS estimator ( ϱ n, δ n ) of (ϱ, δ) based on a sample X,...,X n can be obtained by minimizing the sum of squares X ] ϱ + δ ϱ δ X ϱ δ ϱ+ δ m ε with respect to (ϱ, δ) over R, and it has the form n, x m ε, x ϱ n := n, x, (3.) δ n := n ũ, x m ε ũ, x n ũ, x (3.) on the set H n H n, where := ] ] R, ũ := R, and { H n := (x,...,x n ) ( } R ) n :, x >, (3.3) { H n := (x,...,x n ) ( } R ) n : ũ, x >, (3.4) where x := is the zero vector in R. In a natural way, we extend the CLS estimators ϱ n and δ n to the set H n and H n, respectively. Moreover, for each n N, any CLS estimator ( α n, β n ) of the offspring means (α, β) based on a sample X,...,X n has the form αn β n ] = ] ] ϱn, (3.5) δ n whenever the sample belongs to the set H n H n. For the proof see Ispány et al. 8], Lemma A.. In what follows, we always assume that (X ) Z+ is a -type doubly symmetric Galton Watson process with offspring means (α, β) (, ) such that α + β = (hence it is critical and positively regular), X =, E(,, 8 )<, E(,, 8 )<, E( ε 8 )<, and m ε. Then lim n ((X,...,X n ) H n ) =. If V ũ, ũ >, or if V ũ, ũ = and E( ũ, ε )>, then lim n ((X,...,X n ) H n ) =, see roposition A.3.

5 -type critical Galton Watson processes with immigration 5 Let (Y t ) t R+ be the unique strong solution of the stochastic differential equation (SDE) dy t =, m ε dt + V, Y t + dw t, t R +, Y =, (3.6) where (W t ) t R+ is a standard Wiener process. Theorem 3.. We have n( ϱ n ) D Y t d(y t, m ε t), as n. (3.7) Y t dt If V, =, then ( n 3/ ( ϱ n ) D N, 3 V ) ε,, m ε, as n. (3.8) If V ũ, ũ >, then n / ] ( α n α) D n / αβ Y t d W t ( β n β) Y t dt ], as n, (3.9) where ( W t ) t R+ is a standard Wiener process, independent from (W t ) t R+. If V ũ, ũ = and E( ũ, ε )>, then n / ] ) ] ( α n α) D V ε ũ, ũ n / N (, ( β n β) 4E( ũ, ε, as n. (3.) ) Remar 3.. If V ũ, ũ > and V, = then in (3.9)wehave αβ Y ] ( t d W t D Y = N, 4 ) ] t dt 3 αβ. a.s. Remar 3.3. Note that the assumption V, = is fulfilled if and only if,,, +,,, = a.s. and,,, +,,, =, that is, the total number of offsprings produced by an individual of type is, and the same holds for individuals of type. In a similar way, the assumption V ũ, ũ = is fulfilled if and only if α = β =,,,, a.s. a.s. =,,, and,,, =,,,, that is, the number of offsprings of type and of type produced by an individual of type are the same, and the same holds for individuals of type. Observe that the assumptions V, = and V ũ, ũ = can not be fulfilled at the same time. Condition E( ũ, ε a.s. )> fails to hold if and only if ε, ε, =, and, under the assumption V ũ, ũ =, this implies X, = X, (see Lemma A.), when ((X,...,X n ) a.s. H n H n ) = for all n N, and hence the LSE of the offspring means (α, β) is not defined uniquely, see Appendix A.

6 5 M. Ispány, K. Körmendi and G. ap Remar 3.4. For each n N, consider the random step process Theorem 5. implies convergence (5.3), hence X (n) X (n) t := n X nt, t R +. D X := Y as n, (3.) where the process (Y t ) t R+ is the unique strong solution of the SDE (3.6) with initial value Y =. Note that convergence (3.) holds even if V, =, when the unique strong solution of (3.6) is the deterministic function Y t =, m ε t, t R +. The SDE (3.6) has a unique strong solution (Y (y) t ) t R+ for all initial values Y (y) = y R, and if y, then Y t (y) is nonnegative for all t R + with probability one, hence Y t + may be replaced by Y t under the square root in (3.6), see, for example, Barczy et al. 3], Remar 3.3. Remar 3.5. We note that in the critical positively regular case the limit distributions for the CLS estimators of the offspring means (α, β) are concentrated on the line {(u, v) R : u + v = }.In order to handle the difficulty caused by this degeneracy, we use an appropriate reparametrization. Surprisingly, the scaling factor of the CLS estimators of (α, β) is always n, which is the same as in the subcritical case. The reason of this strange phenomenon can be understood from the joint asymptotic behavior of the numerator and the denominator of the CLS estimators given in Theorems 4., 4. and 4.3. The scaling factor of the estimators of the criticality parameter ϱ is usually n, except in a particular special case of V, =, when it is n 3/. One of the decisive tools in deriving the needed asymptotic behavior is a good bound for the moments of the involved processes, see Corollary B.6. Remar 3.6. The shape of Y t d(y t, m ε t)/ Y t dt in (3.7) is similar to the limit distribution of the Dicey Fuller statistics for unit root test of AR() time series, see, for exmple, Hamilton 6], formulas 7.4. and 7.4.7, or Tanaa 9], (7.4) and Theorem The shape of Y t d W t / Y t dt in (3.9) is also similar, but it contains two independent standard Wiener processes. This phenomenon is very similar to the appearance of two independent standard Wiener processes in limit theorems for CLS estimators of the variance of the offspring and immigration distributions for critical branching processes with immigration in Winnici ], Theorems 3.5 and 3.8. Finally, note that the limit distribution of the CLS estimator of the criticality parameter ϱ is non-symmetric and non-normal in (3.7), and symmetric normal in (3.8), but the limit distribution of the CLS estimator of the offspring means (α, β) is always symmetric, although non-normal in (3.9). Remar 3.7. The eighth order moment conditions on the offspring and immigration distributions in Theorem 3. seem to be too strong, but we note that the process (X ) Z+ can be considered as a heteroscedastic time series. Indeed, X = m X + m ε + M,see(4.), and by (B.), E(M M F ) = X, V + X, V + V ε, N. That is why we thin that the behavior of the process (X ) Z+ is similar to GARCH models, where, even in the stable case, high moment conditions are needed for convergence of estimators such as the quasi-maximum lielihood estimator in Hall and Yao 5] or the Whittle estimator in Miosch and Straumann 4].

7 -type critical Galton Watson processes with immigration roof of the main results Applying (.), let us introduce the sequence M := X E(X F ) = X m X m ε, N, (4.) of martingale differences with respect to the filtration (F ) Z+.By(4.), the process (X ) Z+ satisfies the recursion Next, let us introduce the sequence X = m X + m ε + M, N. (4.) U :=, X =X, + X,, Z +. One can observe that U for all Z +, and U = U +, m ε +, M, N, (4.3) since, m X = m X = X = U, because ϱ = α + β = implies that is a left eigenvector of the mean matrix m belonging to the eigenvalue. Hence, (U ) Z+ is a nonnegative unstable AR() process with positive drift, m ε and with heteroscedastic innovation (, M ) N. Moreover, let Note that we have V := ũ, X =X, X,, Z +. V = (α β)v + ũ, m ε + ũ, M, N, (4.4) since ũ, m X =ũ m X = (α β)ũ X = (α β)v, because ũ is a left eigenvector of the mean matrix m belonging to the eigenvalue α β. Thus (V ) N is a stable AR() process with drift ũ, m ε and with heteroscedastic innovation ( ũ, M ) N. Observe that X, = (U + V )/, X, = (U V )/, Z +. (4.5) By (3.), for each n N,wehave ϱ n = n, M U n U, whenever (X,...,X n ) H n, where H n, n N, aregivenin(3.3). By (3.), for each n N, we have n ũ, M V δ n δ = n V, (4.6) whenever (X,...,X n ) H n, where H n, n N,aregivenin(3.4). Theorem 3. will follow from the following statements by the continuous mapping theorem.

8 54 M. Ispány, K. Körmendi and G. ap Theorem 4.. We have, as n, Y n 3 U t dt n V D () V ũ, ũ n, M U Y t dt Y t d ( Y t, m ε t ). n 3/ ũ, M V () / V ũ, ũ Y t d W t Theorem 4.. If V, = then, as n, n 3 U n V n 3/, M U n 3/ ũ, M V Y t dt D () V ũ, ũ Y t dt V ε, /, Y t d W t () / V ũ, ũ Y t d W t where ( W t ) t R+ is a standard Wiener process, independent from (W t ) t R+ and ( W t ) t R+. Note that (Y t ) t R+ is now the deterministic function Y t =, m ε t, t R +, hence Y t dt =, m ε /3, Y t dt =, m ε /, Y t d W t =, m ε t d W t and Y t d W t =, m ε t d W t. Theorem 4.3. If V ũ, ũ = then, as n, n 3 U Y t dt n V D E ( ũ, ε n, M U ) n / ũ, M V Y t d ( Y t, m ε t ). Vε ũ, ũ E ( ũ, ε )] / W 5. roof of Theorem 4. Consider the sequence of stochastic processes t := Z (n) M (n) t N (n) t (n) t := Z (n),

9 -type critical Galton Watson processes with immigration 55 with Z (n) := n ] M n M U n 3/ M V n ] = n U n 3/ V M for t R + and,n N, where denotes Kronecer product of matrices. Theorem 4. follows from Lemma A. and the following theorem (this will be explained after Theorem 5.). Theorem 5.. We have Z (n) D Z, as n, (5.) where the process (Z t ) t R+ with values in (R ) 3 is the unique strong solution of the SDE ] dwt dz t = γ(t,z t ), t R d W +, (5.) t with initial value Z =, where (W t ) t R+ and ( W t ) t R+ are independent -dimensional standard Wiener processes, and γ : R + (R ) 3 (R ) 3 is defined by,(x + tm ε ) + / / V γ(t,x) :=,(x + tm ε ) + 3/ / V ( ) V ũ, ũ /, x + tm ε V / for t R + and x = (x, x, x 3 ) (R ) 3. (Note that the statement of Theorem 5. holds even if V ũ, ũ =, when the last - dimensional coordinate process of the unique strong solution (Z t ) t R+ is.) The SDE (5.) has the form ],(Mt + tm ε ) + / / V dw t dmt dz t = dn t =,(Mt + tm ε ) + 3/ / V dw t ( ) d t V ũ, ũ /, M t + tm ε V /, t R +. d W t Ispány and ap 9] proved that the first -dimensional equation of this SDE has a unique strong solution (M t ) t R+ with initial value M =, and (M t + tm ε ) + may be replaced by M t + tm ε (see the proof of 9, Theorem 3.]). Thus, the SDE (5.) has a unique strong solution with initial value Z =, and we have t, M ] t + tm ε / V / dw s Z t = Mt N t t t =, M t + tm ε dm s ( ) V ũ, ũ / t, M t + tm ε V /, t R +. d W s

10 56 M. Ispány, K. Körmendi and G. ap By the method of the proof of X (n) Lemma C., one can easily derive X (n) where Z (n) D X in Theorem 3. in Barczy et al. 3], applying ] ] D X, as n, (5.3) Z X (n) t := n X nt, X t :=, M t + tm ε, t R +,n N, see Ispány et al. 8], page. Now, with the process we have Y t :=, X t =, M t + tm ε, t R +, X t = Y t, t R +. By Itô s formula, we obtain that the process (Y t ) t R+ satisfies the SDE (3.6). Next, similarly to the proof of (A.), by Lemma C.3, convergence (5.3) and Lemma A. with U =, X implies, X t dt n 3 U V n V ũ, ũ D, X t dt n, M U, n 3/ ũ, M V Y t d, M t ( ) V ũ, ũ / Y t d ũ, V / W t as n. This limiting random vector can be written in the form as given in Theorem 4., since, X t =Y t,, M t =, X t, m ε t = Y t, m ε t and ũ, V / W t = V ũ, ũ / W t for all t R + with a (one-dimensional) standard Wiener process ( W t ) t R+. roof of Theorem 5.. In order to show convergence Z (n) D Z, we apply Theorem D. with the special choices U := Z, U (n) := Z (n) (n), n, N, (F ) Z+ := (F ) Z+ and the function γ which is defined in Theorem 5.. Note that the discussion after Theorem 5. shows that the SDE (5.) admits a unique strong solution (Zt z) t R + for all initial values Z z = z (R ) 3. Now we show that conditions (i) and (ii) of Theorem D. hold. The conditional variance E(Z (n) (Z(n) ) F ) has the form n n 3 U n 5/ V n 3 U n 4 U n 7/ U V V M n 5/ V n 7/ U V n 3 V

11 -type critical Galton Watson processes with immigration 57 for n N, {,...,n}, with V M := E(M M F ), and γ(s,z (n) s (n) (n), M s + sm ε, M s + sm ε (n) (n) 3, M s + sm ε, M s + sm ε V ũ, ũ for s R +, where we used that, M (n) s by (4.), we get + sm ε + =, M (n) s, M (n) s + sm ε ns, M (n) s + sm ε =, X m X m ε +,sm ε n )γ (s, Z s (n) ) has the form V + sm ε, s R +, n N. Indeed, = n, X ns ns ns +, m ε (5.4) n = n U ns ns ns +, m ε R + n for s R +, n N, since m = implies, m X = m X = X =, X. In order to chec condition (i) of Theorem D., we need to prove that for each T>, as n, t,t ] t,t ] V M t,t ] n 3 n U V M U V M t,t ] n 3 n 4 V V M V ũ, ũ t,t ] t t t t, M (n) s, M (n) s, M (n) s, M (n) s t,t ] n 7/ + sm ε V ds, (5.5) V + sm ε ds, (5.6) 3V + sm ε ds, (5.7) V + sm ε ds, (5.8) n 5/ V V M, (5.9) U V V M. (5.)

12 58 M. Ispány, K. Körmendi and G. ap First, we show (5.5). By (5.4), t, M(n) s + sm ε ds has the form nt n U + Using Lemma B., we obtain Thus, in order to show (5.5), it suffices to prove nt nt nt +(nt nt ) n U nt + n, m ε. V M = U V + V (V V ) + V ε. (5.) nt n n t,t ] V t,t ], n t,t ] U nt, (5.) nt + ( nt nt ) ], (5.3) as n.using(b.4) with (l,i,j) = (,, ) and (B.5) with (l,i,j) = (,, ), wehave (5.). Clearly, (5.3) follows from nt nt, n N, t R +, thus we conclude (5.5). The convergences (5.6) and (5.7) can be checed in a similar way. Next,weturntoprove(5.8). By (5.) and (B.4), we get n 3 U V M U V, (5.4) as n for all T >. Using (5.6), in order to prove (5.8), it is sufficient to show that n 3 V V M V nt ũ, ũ U V, (5.5) t,t ] as n for all T>. By (5.), nt V V M has the form nt U V V + (V V ) + V V ε. Using (B.4) with (l,i,j)= (6,, 3) and (l,i,j)= (4,, ), wehave nt n 3 V 3, t,t ] V 3 nt n 3 V, hence (5.5) will follow from n 3 U V V ũ, ũ U as n,, (5.6)

13 -type critical Galton Watson processes with immigration 59 as n for all T>. By the method of the proof of Lemma A., we obtain a decomposition of nt U V as a sum of a martingale and some negligible terms, namely, nt U V = nt U V E( U V = + V ũ, ũ U = + lin. comb. of U V, = F )] (α β) U nt V nt + O(n) V, = = nt U and V. = Using (B.6) with (l,i,j)= (8,, ) we have n 3 t,t ] = U V E( U V F )], as n. Thus, in order to show (5.6), it suffices to prove n 3 nt n 3 t,t ] U V, nt n 3 U U nt V nt, nt n 3 nt n 3 V, n 3/ U nt t,t ], (5.7) V, (5.8), (5.9) as n.using(b.4) with (l,i,j) = (,, ), (l,i,j) = (4,, ), (l,i,j) = (,, ) and (l,i,j)= (,, ), wehave(5.7) and (5.8). By (B.5) with (l,i,j)= (4,, ) and by (B.5), we have (5.9). Thus, we conclude (5.8). Convergences (5.9) and (5.) can be proved similarly. Finally, we chec condition (ii) of Theorem D., that is, the conditional Lindeberg condition nt E ( (n) Z { Z (n) for all θ> and T>. We have E( Z (n) Z (n) >θ} F { Z (n) ), as n (5.) >θ} F ) θ E( Z (n) 4 F ) and 4 3 ( n 4 + n 8 U 4 + n 6 V ) 4 M 4.

14 6 M. Ispány, K. Körmendi and G. ap Hence, for all θ> and T>, we have nt E ( Z (n) ) (n) { Z, as n, >θ} since E( M 4 ) = O( ), E( M 4 U E( M 4 ) 8 )E(U 8 ) = O(6 ) and E( M 4 V 4 ) E( M 8 )E(V 8 ) = O(4 ) by Corollary B.6. Here we call the attention that our eighth order moment conditions E(,, 8 )<, E(,, 8 )< and E( ε 8 )< are used for applying Corollary B.6. This yields (5.). 6. roof of Theorem 4. This is similar to the proof of Theorem 4.. Consider the sequence of stochastic processes t := Z (n) M (n) t N (n) t (n) t := Z (n) with Z (n) := n ] M n 3/, M U n 3/ M V for t R + and,n N. Theorem 4. follows from Lemma A. and the following theorem (this will be explained after Theorem 6.). Theorem 6.. If V, = then Z (n) D Z, as n, (6.) where the process (Z t ) t R+ with values in R R R is the unique strong solution of the SDE dw t dz t = γ(t,z t ) d W t, t R +, (6.) d W t with initial value Z =, where (W t ) t R+, ( W t ) t R+ and ( W t ) t R+ are independent standard Wiener processes of dimension, and, respectively, and γ(t,x) is a bloc diagonal matrix with the matrices,(x + tm ε ) + / V /, V ε, /, m ε t and ( V ũ,ũ )/, x + tm ε V / in its diagonal for each t R + and x = (x,x, x 3 ) R R R.

15 -type critical Galton Watson processes with immigration 6 As in the case of Theorem 4., the SDE (6.) has a unique strong solution with initial value Z =,forwhichwehave Z t = Mt N t t t Y / ] t V / dw s t = V ε, /, m ε s d W s, t R +, ( ) V ũ, ũ / t Y t V / d W s where now V, = yields Y t =, m ε t, t R +. One can again easily derive where X (n) Z (n) ] ] D X, as n, (6.3) Z X (n) t := n X nt, X t :=, M t + tm ε = t, m ε, for t R + and n N, since X t = Y t = t, m ε, t R +. Next, similarly to the proof of (A.), by Lemma C.3, convergence (6.3) and Lemma A. with U =, X imply n 3 U n V n 3/, M U n 3/ ũ, M V, X t dt ( ) V ũ, ũ D, X t dt, V ε, /, m ε t d W t ( ) V ũ, ũ / Y t d ũ, V / W t as n. This limiting random vector can be written in the form as given in Theorem 4. since, X t =Y t =, m ε t, and ũ, V / W t = V ũ, ũ / W t for all t R + with a (onedimensional) standard Wiener process ( W t ) t R+. roof of Theorem 6.. Similar to the proof of Theorem 5.. The conditional variance ) F ) has the form E(Z (n) (Z(n) n V M n 5/ U V M n 5/ V V M n 5/ U V M n 3 U V M n 3 U V V M n 5/ V V M n 3 U V V M n 3 V M

16 6 M. Ispány, K. Körmendi and G. ap for n N, {,...,n}, with V M := E(M M F ), and γ(s,z s (n) )γ (s, Z s (n) ) has the form, M (n) s + sm ε V V ε,, m ε s V ũ, ũ, M (n) V s + sm ε for s R +. In order to chec condition (i) of Theorem D., we need to prove only that for each T>, t,t ] n 3 t,t ] U V M t,t ] t n 5/ U V M, (6.4) V ε,, m ε s ds, (6.5) U V V M, (6.6) n 3 as n, since the rest, namely, (5.5), (5.8) and (5.9) have already been proved. Clearly, V, = implies V, = and V, =. For each i {, }, wehave V i, = V i = (V / ) (V / i ) = V / i, hence we obtain V / i =, thus V i i = V / (V / i ) =, and hence V i i =, implying also V =. First we show (6.4). By (5.), V = and V i = for i {, }, we obtain nt U V M = U V ε, (6.7) hence using (B.4) with (l,i,j)= (,, ), we conclude (6.4). Now we turn to chec (6.5). By (5.), nt U V M = U V ε = U V ε,, hence, in order to show (6.5), it suffices to prove U t3 3, m ε, as n. (6.8) t,t ] n 3

17 -type critical Galton Watson processes with immigration 63 We have where U t3 3, m ε U n 3 ( ), m ε n 3 t,t ] n 3 + ( ) t3 3, m ε, n 3 ( ) t3 3, hence, in order to show (6.5), it suffices to prove n 3 nt as n, U, m ε, as n. (6.9) For all N, by Remar 3.3, V, = implies U = X, j= X, (,j,, +,j,, ) + j= (,j,, +,j,, ) + (ε, + ε, ) a.s. = X, + X, + ε, + ε, = U +, ε, hence U = i=, ε i. By Kolmogorov s maximal inequality, ( n max U, m ε ) ε n ε Var(U nt ) {,..., nt } as n for all ε>, thus We have hence n max {,..., nt } n max {,..., nt } U, m ε, = nt n ε Var(, ε ) as n. U, m ε U, m ε +, m ε U, m ε, U, m ε (n max {,..., nt } U, m ε ) + nt n, m ε max {,..., nt } U, m ε,

18 64 M. Ispány, K. Körmendi and G. ap as n. Consequently, n 3 nt U ( ), m ε nt n 3 max {,..., nt } as n, thus we conclude (6.9), and hence (6.5). Finally, we chec (6.6). By (5.), nt U ( ), m ε, U V V M = U V V ε, hence using (B.4) with (l,i,j)= (,, ), we conclude (6.6). Condition (ii) of Theorem D. can be checed as in case of Theorem roof of Theorem 4.3 This proof is also similar to the proof of Theorem 4.. Consider the sequence of stochastic processes M (n) Z t (n) t := N t (n) n ] M := Z (n) t (n) with Z (n) := n M U n / ũ, M V for t R + and,n N. Theorem 4.3 follows from Lemma A. and the following theorem (this will be explained after Theorem 7.). Theorem 7.. If V ũ, ũ = then Z (n) D Z, as n, (7.) where the process (Z t ) t R+ with values in R R R is the unique strong solution of the SDE ] dwt dz t = γ(t,z t ), t R d W +, (7.) t with initial value Z =, where (W t ) t R+ and ( W t ) t R+ are independent standard Wiener processes of dimension and, respectively, and γ : R + (R R R) R 5 3 is defined by,(x + tm ε ) + / / V γ(t,x) :=,(x + tm ε ) + 3/ / V Vε ũ, ũ E ( ũ, ε )] /

19 -type critical Galton Watson processes with immigration 65 for t R + and x = (x, x,x 3 ) R R R. As in the case of Theorem 4., the SDE (7.) has a unique strong solution with initial value Z =,forwhichwehave t ] Ys / V / Mt dw s Z t = N t = t t Y s dm s Vε ũ, ũ E (, t R +. ũ, ε )] / W t One can again easily derive X (n) Z (n) ] ] D X, as n, (7.3) Z where X (n) t := n X nt, X t :=, M t + tm ε, t R +,n N. Next, similarly to the proof of (A.), by Lemma C.3, convergence (7.3) and Lemma A. imply n 3 U, X t dt n V D E ( ũ, ε n, M U ), n / ũ, M V Y t d, M t Vε ũ, ũ E ( ũ, ε )] / W as n. Note that this convergence holds even in case E ũ, ε ]=. The limiting random vector can be written in the form as given in Theorem 4.3, since, X t =Y t and, M t = Y t, m ε t for all t R +. roof of Theorem 7.. Similar to the proof of Theorem 5.. The conditional variance E(Z (n) (Z(n) ) F ) has the form n V M n 3 U V M n 3/ V V M ũ n 3 U V M n 4 U V M n 5/ U V V M ũ n 3/ V ũ V M n 5/ U V ũ V M n V V M ũ for n N, {,...,n}, with V M := E(M M F ), and γ(s,z (n) s )γ (s, Z s (n) ) has the form (n) (n) V, M s + sm ε V, M s + sm ε, M (n) V s + sm ε, M (n) 3V s + sm ε V ε ũ, ũ E ( ũ, ε )

20 66 M. Ispány, K. Körmendi and G. ap for s R +. In order to chec condition (i) of Theorem D., we need to prove only that for each T>, t,t ] n ũ V M ũ t V ε ũ, ũ E ( ũ, ε ), (7.4) V t,t ] t,t ] n 5/ n 3/ V ũ V M, (7.5) U V ũ V M, (7.6) as n, since the rest, namely, (5.5), (5.6) and (5.7), have already been proved. Clearly, V ũ, ũ = implies V ũ, ũ = and V ũ, ũ =. For each i {, }, wehave V i ũ, ũ =ũ V i ũ = (V / ũ) (V / i ũ) = V / i ũ, hence we obtain V / i ũ =, thus V i i ũ = V / (V / i ũ) =, and hence ũ V i i =. First, we show (7.4). By (5.), V ũ V M ũ = hence, in order to show (7.4), it suffices to prove t,t ] n V For all N, by Remar 3.3, V ũ, ũ = implies V = X, j= X, (,j,,,j,, ) + a.s. = ε, ε, = ũ, ε. V ũ V ε ũ, te( ũ, ε ). j= (,j,,,j,, ) + (ε, ε, ) We have n te( ũ, ε ) ũ, ε E ( ũ, ε )] n V + nt nt E ( ũ, ε ), n

21 -type critical Galton Watson processes with immigration 67 where nt nt, hence, in order to show (7.4), it suffices to prove n t,t ] = n ũ, ε E ( ũ, ε )] max N {,..., nt } N ũ, ε E ( ũ, ε )]. (7.7) Applying Kolmogorov s maximal inequality, we obtain ( n max N {,..., nt } ( nt n ε Var N ũ, ε E ( ) ũ, ε )] ε ũ, ε ) = nt n ε Var( ũ, ε ), as n for all ε>, thus we conclude (7.7), and hence (7.4). Now we turn to chec (7.5). By (5.), nt V ũ V M = V ũ V ε. Again by the strong law of large numbers, n nt T>, hence we conclude (7.5). Finally, we chec (7.6). By (5.), V a.s. U V ũ E ( M M F ) = U V ũ V ε. te( ũ, ε ) as n for all Applying V = ũ, ε, N, and Corollary B.6, we have E( U V ) E(U )E(V ) = O(), which clearly implies (7.6). Condition (ii) of Theorem D. can be checed again as in case of Theorem 5.. Appendix A: CLS estimators In order to analyse existence and uniqueness of the estimators given in (3.), (3.) and (3.5) in case of a critical doubly symmetric -type Galton Watson process, that is, when ϱ =, we need the following approximations.

22 68 M. Ispány, K. Körmendi and G. ap Lemma A.. We have n ( V V ũ, ũ ) U, as n. roof. In order to prove the statement, we derive a decomposition of n V as a sum of a martingale and some negligible terms. Using recursion (4.4), Lemma B. and (4.5), we obtain Thus, E ( V F ) = (α β) V + (α β) ũ, m ε V + ũ, m ε + ũ E ( M M F )ũ Consequently, V = (α β) V + ũ (V + V )ũu + constant + constant V. = V E ( V F + O(n) + constant )] + (α β) V. V + ũ V ũ U V = (α β) + V E ( V (α β) V ũ, ũ F U (α β) (α β) V n + O(n) + constant V. )] (A.) Using (B.6) with (l,i,j)= (8,, ), we obtain n V E ( V F )], as n. By Corollary B.6, we obtain E(Vn ) = O(n), and hence n Vn. Moreover, n n V asn follows by (B.4) with the choices (l,i,j) = (4,, ). Consequently, by (A.), we obtain the statement, since (α β) =.

23 -type critical Galton Watson processes with immigration 69 Lemma A.. If V ũ, ũ =, then n V a.s. E ( ũ, ε ), as n, and E( ũ, ε ) = if and only if X, a.s. = X, for all N. a.s. roof. By Remar 3.3, V ũ, ũ = implies V = ε, ε, = ũ, ε for all N, hence the convergence follows from the strong law of large numbers. Clearly E( ũ, ε ) = is equivalent a.s. a.s. to ũ, ε =ε, ε, =, and hence it is equivalent to X, X, = for all N. Now we can prove existence and uniqueness of CLS estimators of the offspring means and of the criticality parameter. roposition A.3. We have lim n ((X,...,X n ) H n ) =, where H n is defined in (3.3), and hence the probability of the existence of a unique CLS estimator ϱ n converges to as n, and this CLS estimator has the form given in (3.) whenever the sample (X,...,X n ) belongs to the set H n. If V ũ, ũ >, or if V ũ, ũ = and E( ũ, ε )>, then lim n ((X,...,X n ) H n ) =, where H n is defined in (3.4), and hence the probability of the existence of unique CLS estimators δ n and ( α n, β n ) converges to as n. The CLS estimator δ n has the form given in (3.) whenever the sample (X,...,X n ) belongs to the set H n. The CLS estimator ( α n, β n ) has the form given in (3.5) whenever the sample (X,...,X n ) belongs to the set H n H n. roof. Recall convergence X (n) show ( X n 3, + X,) D D X = Y from (3.). By Lemmas C. and C.3 one can Y t dt, as n, (A.) see Ispány et al. 8], roposition A.4. Since m ε, by the SDE (3.6), we have (Y t =,t, ]) =, which implies that ( Y t dt >) =. Consequently, the distribution function of Y t dt is continuous at, and hence, by (A.), ( ) ( ), X > Yt dt > =, as n. Now pose that V ũ, ũ > holds. In a similar way, using Lemma A., convergence (3.), and Lemmas C. and C.3, one can show n ũ, X D V ũ, ũ Y t dt, as n,

24 7 M. Ispány, K. Körmendi and G. ap implying ( ) ũ, X > ( ) Y t dt> =, as n, hence we obtain the statement under the assumption V ũ, ũ >. Next, we pose that V ũ, ũ = and E( ũ, ε )> hold. Then ( ) ( ) ũ, X > = V n >, as n, since Lemma A. yields n n V E( ũ, ε )>, and hence we conclude the statement under the assumptions V ũ, ũ = and E( ũ, ε )>. Appendix B: Estimations of moments In the proof of Theorem 3., good bounds for moments of the random vectors and variables (M ) Z+, (X ) Z+, (U ) Z+ and (V ) Z+ are extensively used. First note that, for all N, E(M F ) = and E(M ) =, since M = X E(X F ). Lemma B.. Let (X ) Z+ be a -type Galton Watson process with immigration and with X =. If E(,, )<, E(,, )< and E( ε )< then E ( M M F ) = X, V + X, V + V ε, N. (B.) If E(,, 3 )<, E(,, 3 )< and E( ε 3 )<, then E ( M 3 F ) = X, E (,, E(,, ) 3] (B.) + X, E (,, E(,, ) 3] + E (ε E(ε ) 3], N. roof. By (.) and (4.), M has the form X, (,j, E(,j, ) ) + X, (,j, E(,j, ) ) + ( ε E(ε ) ) (B.3) j= j= for all N. The random vectors {,j, E(,j, ),,j. E(,j, ), ε E(ε ) : j N} are independent of each other, independent of F, and have zero mean vector, thus we conclude (B.) and (B.). Lemma B.. Let (ζ ) N be independent and identically distributed random vectors with values in R d such that E( ζ l )< with some l N.

25 -type critical Galton Watson processes with immigration 7 (i) Then there exists Q = (Q,...,Q d l) : R R dl, where Q,...,Q d l having degree at most l such that are polynomials E ( (ζ + +ζ N ) l) = N l E(ζ ) ] l + Q(N), N N,N l. (ii) If E(ζ ) =, then there exists R = (R,...,R d l) : R R dl, where R,...,R d l are polynomials having degree at most l/ such that E ( (ζ + +ζ N ) l) = R(N), N N,N l. The coefficients of the polynomials Q and R depend on the moments E(ζ i ζ il ), i,...,i l {,...,N}. roof. (i) We have E ( (ζ + +ζ N ) l) = s {,...,l},,..., s Z +, + + +s s =l, s (i,...,i l ) (N,l),...,s ( )( ) ( ) N N N s E(ζ i ζ il ), where the set (N,l),..., s consists of permutations of all the multisets containing pairwise different elements j,...,j s of the set {,...,N} with multiplicities,..., s, respectively. Since ( )( ) ( ) N N N s = N(N ) (N s + )!! s! is a polynomial of the variable N having degree + + s l, there exists = (,..., d l) : R R dl, where,..., d l are polynomials having degree at most l such that E((ζ + +ζ N ) l ) = (N). A term of degree l can occur only in case + + s = l, when + + +s s = l implies s = and = l, thus the corresponding term of degree l is N(N ) (N l + )E(ζ )] l, hence we obtain the statement. art (ii) can be proved in a similar way. Lemma B. can be generalized in the following way. Lemma B.3. For each i N, let (ζ i, ) N be independent and identically distributed random vectors with values in R d such that E( ζ i, l )< with some l N. Let j,...,j l N. s s

26 7 M. Ispány, K. Körmendi and G. ap (i) Then there exists Q = (Q,...,Q d l) : R l R dl, where Q,...,Q d l are polynomials of l variables having degree at most l such that E ( (ζ j, + +ζ j,n ) (ζ jl, + +ζ jl,n l ) ) = N N l E(ζ j,) E(ζ jl,) + Q(N,...,N l ) for N,...,N l N with N l,...,n l l. (ii) If E(ζ j,) = =E(ζ jl,) =, then there exists R = (R,...,R d l) : R l R dl, where R,...,R d l are polynomials of l variables having degree at most l/ such that E ( (ζ j, + +ζ j,n ) (ζ jl, + +ζ jl,n l ) ) = R(N,...,N l ) for N,...,N l N with N l,...,n l l. The coefficients of the polynomials Q and R depend on the moments E(ζ j,i ζ jl,i l ), i {,...,N },...,i l {,...,N l }. Lemma B.4. If (α, β), ] with α + β =, then the matrix m defined in (.4) has eigenvalues and α β, and the powers of m tae the form m j = ] + ] (α β)j, j Z +. Consequently, m j =O(), that is, j N mj <. Lemma B.5. Let (X ) Z+ be a -type doubly symmetric Galton Watson process with immigration with offspring means (α, β), ] such that α + β = (hence it is critical). Suppose X =, and E(,, l )<, E(,, l )<, E( ε l )< with some l N. Then E( X l ) = O( l ), that is, N l E( X l )<. roof. The statement is clearly equivalent with E( (X,,X, ) ) c l, N, for all polynomials of two variables having degree at most l, where c depends only on. If l =, then (.3) and Lemma B.4 imply E(X ) = j= ( m j m ε = ] (α β) + 4β ]) m ε, for all N, which yields the statement. Using part (i) of Lemma B.3 and separating the terms having degree and less than, we obtain E ( X F ) = X, m + X, m + X, X, (m m + m m ) + Q (X,,X, )

27 -type critical Galton Watson processes with immigration 73 = (X, m + X, m ) + Q (X,,X, ) = (m X ) + Q (X,,X, ) = m X + Q (X,,X, ), where Q = (Q,,Q,,Q,3,Q,4 ) : R R 4, and Q,, Q,, Q,3 and Q,4 are polynomials of two variables having degree at most. Hence In a similar way, E ( X E ( X l ) = m E ( X ) + E Q (X,,X, ) ]. ) = m l E ( X l ) + E Ql (X,,X, ) ], where Q l = (Q l,,...,q l, l) : R R l, and Q l,,...,q l, l are polynomials of two variables having degree at most l, implying E ( X l ) ( = m l) j E Ql (X j,,x j, ) ] j= ( = m l) j E Ql (X j,,x j, ) ] j= ( j ) le = m Ql (X j,,x j, ) ]. j= Let us pose now that the statement holds for,...,l. Then E Ql,i (X j,,x j, ) ] c Ql,i l, N, i {,..., l}. By Lemma B.4 (m j ) l =O(), hence we obtain the assertion for l. Corollary B.6. Let (X ) Z+ be a -type doubly symmetric Galton Watson process with immigration having offspring means (α, β) (, ) such that α + β = (hence it is critical and positively regular). Suppose X =, and E(,, l )<, E(,, l )<, E( ε l )< with some l N. Then E( X l ) = O( l ), E(M l ) = O( l/ ), E(U l) = O(l ) and E(V j ) = O( j ) for j Z + with j l. roof. The first statement is just Lemma B.5. Next, we turn to prove E(M l ) = O( l/ ). Using (B.3), part (ii) of Lemma B.3, and that the random vectors {,j, E(,j, ),,j. E(,j, ), ε E(ε ) : j N} are independent of each other, independent of F, and have zero mean vector, we obtain E(M l F ) = R(X,,X, ) with R = (R,...,R l) : R R l, where R,...,R l are polynomials of two variables having degree at most l/. Hence E(M l ) = E(R(X,,X, )). By Lemma B.5, we conclude E(M l ) = O( l/ ).Therest of the proof can be carried out as in Corollary 9. of Barczy et al. ].

28 74 M. Ispány, K. Körmendi and G. ap The next corollary can be derived as Corollary 9. of Barczy et al. ]. Corollary B.7. Let (X ) Z+ be a -type doubly symmetric Galton Watson process with immigration having offspring means (α, β) (, ) such that α + β = (hence, it is critical and positively regular). Suppose X =, and E(,, l )<, E(,, l )<, E( ε l )< with some l N. Then (i) for all i, j Z + with max{i, j} l/, and for all κ>i+ j +, we have n κ U i V j, as n, (B.4) (ii) for all i, j Z + with max{i, j} l, for all T>, and for all κ>i+ j + i+j l, we have n κ t,t ] U i nt V j nt, as n, (B.5) (iii) for all i, j Z + with max{i, j} l/4, for all T>, and for all κ>i+ j +, we have n κ U i V j E( U i V j F )], as n. (B.6) t,t ] Remar B.8. In the special case (l,i,j) = (,, ), one can improve (B.5), namely, one can show see Barczy et al. ]. n κ U nt t,t ], as n for κ>, (B.7) Appendix C: A version of the continuous mapping theorem A function f : R + R d is called càdlàg if it is right continuous with left limits. Let D(R +, R d ) and C(R +, R d ) denote the space of all R d -valued càdlàg and continuous functions on R +,respectively. Let B(D(R +, R d )) denote the Borel σ -algebra on D(R +, R d ) for the metric defined in Jacod and Shiryaev ], Chapter VI, (.6) (with this metric D(R +, R d ) is a complete and separable metric space and the topology induced by this metric is the so-called Sorohod topology). For R d -valued stochastic processes (Y t ) t R+ and (Y t (n) ) t R+, n N, with càdlàg paths, we write Y (n) D Y if the distribution of Y (n) on the space (D(R +, R), B(D(R +, R d ))) converges wealy to the distribution of Y on the space (D(R +, R), B(D(R +, R d ))) as n. Concerning the notation D we note that if and n, n N, are random elements with values in a metric D space (E, d), then we also denote by n the wea convergence of the distributions of n

29 -type critical Galton Watson processes with immigration 75 on the space (E, B(E)) towards the distribution of on the space (E, B(E)) as n, where B(E) denotes the Borel σ -algebra on E induced by the given metric d. The following version of continuous mapping theorem can be found, for example, in Kallenberg, Theorem 3.7]. Lemma C.. Let (S, d S ) and (T, d T ) be metric spaces and ( n ) n N, be random elements with D values in S such that n as n. Let f : S T and fn : S T, n N, be measurable mappings and C B(S) such that ( C) = and lim n d T (f n (s n ), f (s)) = if lim n d S (s n,s)= and s C. Then f n ( n ) D f(), as n. For the case S = D(R +, R d ) and T = R q (or T = D(R +, R q )), where d, q N, we formulate a consequence of Lemma C.. For functions f and f n, n N,inD(R +, R d lu ), we write f n f if (f n ) n N converges to f locally uniformly, that is, if t,t ] f n (t) f(t) asn for all T>. For measurable mappings : D(R +, R d ) R q (or : D(R +, R d ) D(R +, R q )) and n : D(R +, R d ) R q (or n : D(R +, R d ) D(R +, R q )), n N, we will denote by C,( n ) n N the set of all functions f C(R +, R d ) such that n (f n ) (f ) (or n (f n ) lu lu (f )) whenever f n f with f n D(R +, R d ), n N. We will use the following version of the continuous mapping theorem several times, see, for example, Ispány and ap ], Lemma 3.. Lemma C.. Let d,q N, and (U t ) t R+ and (U t (n) ) t R+, n N, be R d -valued stochastic processes with càdlàg paths such that U (n) D U. Let : D(R+, R d ) R q (or : D(R +, R d ) D(R +, R q )) and n : D(R +, R d ) R q (or n : D(R +, R d ) D(R +, R q )), n N, be measurable mappings such that there exists C C,( n ) n N with C B(D(R +, R d )) and (U C) =. Then n (U (n) ) D (U). In order to apply Lemma C., we will use the following statement several times, see Barczy et al. ], Lemma B.3. Lemma C.3. Let d,p,q N, h : R d R q be a continuous function and K :, ] R d R p be a function such that for all R> there exists C R > such that K(s,x) K(t,y) ( ) C R t s + x y (C.) for all s,t, ] and x,y R d with x R and y R. Moreover, let us define the mappings, n : D(R +, R d ) R q+p, n N, by ( n (f ) := h ( f() ), ( ( ) ( )) ) K n n,f,f, n n ( (f ) := h ( f() ), K ( u, f (u), f (u) ) ) du

30 76 M. Ispány, K. Körmendi and G. ap for all f D(R +, R d ). Then the mappings and n, n N, are measurable, and C,( n ) n N = C(R +, R d ) B(D(R +, R d )). Appendix D: Convergence of random step processes We recall a result about convergence of random step processes towards a diffusion process, see Ispány and ap ]. This result is used for the proof of convergence (5.). Theorem D.. Let γ : R + R d R d r be a continuous function. Assume that uniqueness in the sense of probability law holds for the SDE du t = γ (t, U t ) dw t, t R +, (D.) with initial value U = u for all u R d, where (W t ) t R+ is an r-dimensional standard Wiener process. Let (U t ) t R+ beasolutionof (D.) with initial value U = R d. For each n N, let (U (n) ) N be a sequence of d-dimensional martingale differences with ) Z+, that is, E(U (n) (n) F ) =, n N, N. Let respect to a filtration (F (n) t := U (n) U (n), t R +,n N. Suppose E( U (n) )< for all n, N. Suppose that for each T>, (i) t,t ] nt (ii) nt E( U(n) where E(U(n) (U(n) ) F (n) (n) (n) { U >θ} F ) ) t for all θ>, (n) γ (s, U s )γ (s, U s (n) ) ds, denotes convergence in probability. Then U (n) D U, as n. Note that in (i) of Theorem D., denotes a matrix norm, while in (ii) it denotes a vector norm. Acnowledgements The authors have been partially ported by the Hungarian Chinese Intergovernmental S&T Cooperation rogramme for 3 under Grant No M. Ispány has been partially ported by the TÁMO-4...C-//KONV-- project. The project has been ported by the European Union, co-financed by the European Social Fund. K. Körmendi was ported by the European Union and the State of Hungary, co-financed by the European Social Fund in the framewor of TÁMO A/---- National Excellence rogram. G. ap has been partially ported by the Hungarian Scientific Research Fund under Grant No. OTKA T-798.

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