2. Indecomposable multi-type branching processes with immigration

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URL:http://www.emath.fr/ps/

ASYMPTOTIC BEHAVIOR OF CRITICAL INDECOMPOSABLE MULTI-TYPE BRANCHING PROCESSES WITH IMMIGRATION

^∗

Tivadar Danka

¹

and Gyula Pap

¹

Abstract. In this paper the asymptotic behavior of a critical multi-type branching process with immigration is described when the offspring mean matrix is irreducible, in other words, when the process is indecomposable. It is proved that sequences of appropriately scaled random step functions formed from periodic subsequences of a critical indecomposable multi-type branching process with immigration converge weakly towards a process supported by a ray determined by the Perron vector of the offspring mean matrix. The types can be partitioned into nonempty mutually disjoint subsets (according to communication of types) such that the coordinate processes belonging to the same subset are multiples of the same squared Bessel process, and the coordinate processes belonging to different subsets are independent.

1991 Mathematics Subject Classification. 60J80, 60F17, 60J60.

The dates will be set by the publisher.

1. Introduction

Let (X_k)_k∈_Z₊ be a single-type Galton–Watson branching process with immigration given by

Xk =

Xk−1

X

j=1

ξk,j+εk, k∈N,

and with initial value X0 = 0, where Xk denotes the number of individuals in the k^th generation, ξk,j

denotes the number of offsprings produced by the j^th individual belonging to the (k−1)^th generation, ε_k denotes the number of immigrants in the k^th generation, {ξk,j, ε_k:k, j∈N} are supposed to be independent, {ξk,j :k, j∈N} and {εk:k∈N} are supposed to consist of identically distributed random variables, and Z+

and N denote the set of non-negative integers and positive integers, respectively. Suppose that E(ξ²_1,1)<∞, E(ε²₁)<∞, and mξ :=E(ξ1,1) = 1, i.e., the process is critical. Wei and Winnicki [19] proved a functional limit theorem

X⁽ⁿ⁾−→ X^D as n→ ∞, (1.1)

Keywords and phrases:Critical multi-type branching processes with immigration, squared Bessel processes.

∗The research of T. Danka and G. Pap was realized in the frames of T ´AMOP 4.2.4. A/2-11-1-2012-0001 ’National Excellence Program – Elaborating and operating an inland student and researcher personal support system’. The project was subsidized by the European Union and co-financed by the European Social Fund.

1 Bolyai Institute, University of Szeged, Aradi v´ertan´uk tere 1, 6720 Szeged, Hungary.

c EDP Sciences, SMAI 1999

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with X_t⁽ⁿ⁾:=n⁻¹X_bntc for t∈R+, n∈N, where R+ denotes the set of non-negative real numbers, bxc denotes the integer part of x ∈ R, and (Xt)_t∈R₊ is the pathwise unique strong solution of the stochastic differential equation (SDE)

dX_t=m_εdt+ q

V_ξX_t⁺dW_t, t∈R+,

with initial value X0= 0, where m_ε:=E(ε₁), V_ξ:= Var(ξ_1,1), (Wt)_t∈_R₊ is a standard Wiener process, and x⁺ denotes the positive part of x∈R.

We will investigate a p-type branching process (Xk)k∈Z+ with immigration. For simplicity, we suppose that the initial value is X0 =0. For each k∈Z+ and i∈ {1, . . . , p}, the number of individuals of type i in the k^th generation is denoted by Xk,i. The non-negative integer-valued random variable ξk,j,i,` denotes the number of type ` offsprings produced by the j^th individual who is of type i belonging to the (k−1)^th generation. The number of type i immigrants in the k^th generation will be denoted by εk,i. Consider the random vectors

X_k :=





 X_k,1

... X_k,p





, ξ_k,j,i:=





 ξ_k,j,i,1

... ξ_k,j,i,p





, ε_k :=





 ε_k,1

... ε_k,p





.

Then we have

Xk =

p

X

i=1 Xk−1,i

X

j=1

ξ_k,j,i+εk, k∈N. (1.2)

Here

ξ_k,j,i,εk :k, j∈N, i∈ {1, . . . , p} are supposed to be independent. Moreover,

ξ_k,j,i:k, j ∈N for each i ∈ {1, . . . , p}, and {ε_k : k ∈ N} are supposed to consist of identically distributed vectors. Suppose E(kξ_1,1,ik²)<∞ for all i∈ {1, . . . , p} and E(kε₁k²)<∞. Introduce the notations

mξ_i :=E(ξ_1,1,i)∈R^p+, mξ :=m_ξ

1 · · · m_ξ

p

∈R^p×p+ , mε:=E(ε1)∈R^p+, Vξ_i := Var(ξ_1,1,i)∈R^p×p, Vε:= Var(ε1)∈R^p×p.

Note that many authors define the offspring mean matrix as m^>_ξ. For k∈Z+, let F_k :=σ(X₀,X₁, . . . ,X_k).

By (1.2),

E(X_k| F_k−1) =

p

X

i=1

X_k−1,im_ξ_i+m_ε=m_ξX_k−1+m_ε. (1.3)

Consequently, E(Xk) =mξE(Xk−1) +mε, which implies

E(Xk) =

k−1

X

j=0

m^j_ξmε, k∈N. (1.4)

Hence the offspring mean matrix mξ plays a crucial role in the asymptotic behavior of the sequence (Xk)_k∈Z₊. A multi-type branching process (Xk)_k∈Z₊ is referred to respectively as subcritical, critical or supercritical if

%(mξ)<1, %(mξ) = 1 or %(mξ)>1, where %(mξ) denotes the spectral radius of the offspring mean matrix mξ (see, e.g., Athreya and Ney [1] or Quine [17]). The process (Xk)k∈Z+ is called indecomposable if the matrix mξ is irreducible. Note that the matrix mξ is irreducible if and only if for all i, j∈ {1, . . . , p} there exists ni,j ∈N such that the matrix entry (mⁿ_ξ^i,j)i,j is positive. In other words, a process is indecomposable if and only if each type of individual may have progeny of any other type. An indecomposable process (X_k)_k∈_Z₊ is called positively regular if the matrix m_ξ is primitive. Note that the matrix m_ξ is primitive if and only if there exists n∈N such that the matrix entry (mⁿ_ξ)i,j is positive. Note that a critical single-type branching process is positively regular.

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Joffe and M´etivier [11, Theorem 4.3.1] studied positively regular critical multi-type branching processes. They determined the limiting behavior of the martingale part M⁽ⁿ⁾, n∈N, given by M⁽ⁿ⁾_t :=n⁻¹Pbntc

k=1Mk, t∈R⁺, with Mk :=Xk−E(Xk| F_k−1), k∈N (which is a special case of Theorem 3.4).

The result (1.1) has been generalized by Isp´any and Pap [9] for positively regular criticalp-type branching processes with immigration. They proved that

X⁽ⁿ⁾−→ X^D u as n→ ∞,

where X⁽ⁿ⁾_t :=n⁻¹X_bntc for t∈R+, n∈N, the process (Xt)_t∈R₊ is the unique strong solution of the SDE dX_t=v^>m_εdt+

q

v^>(uV_ξ)vX_t⁺dW_t, t∈R+,

with initial value X0 = 0, where u and v denotes the right and left Perron vectors of mξ, and uVξ:=Pp

i=1uiVξ_i, where u= (ui)i∈{1,...,p}.

The aim of the present paper is to obtain a generalization of this result for indecomposable critical multi-type branching processes with immigration. Then the types {1, . . . , p} can be partitioned according to communication of types, namely, into r nonempty mutually disjoint subsets D₁, . . . , D_r such that an individual of type j may not have offspring of type i unless there exists `∈ {1, . . . , r} with i ∈D_`−1 and j ∈D_`, where subscripts are considered modulo r. This partitioning is unique up to cyclic permutation of the subsets. The number r is called the index of cyclicity (in other words, the index of imprimivity) of the matrix m_ξ. Note that r = 1 if and only if the matrix mξ is primitive, i.e., the process is positively regular. We succeeded to determine the joint asymptotic behavior of sequences (nr)⁻¹X_rbntc+i−1

t∈R+, n∈N, i∈ {1, . . . , r}, of random step functions as n → ∞, see Theorem 3.1. It turns out that the limiting diffusion process has the form (m^r−i+1_ξ Yt)t∈R+, i∈ {1, . . . , r}, where the distribution of the process (m^j_ξYt)t∈R+ is the same for all j∈N. Moreover, the process (Yt)_t∈R₊ is 1-dimensional in the sense that for all t∈R+, the distribution of Yt is concentrated on the ray R+·u, where u is the Perron eigenvector of the offspring mean matrix mξ. In fact, partitioning the coordinates of the limit process Yt= (Yt,1, . . . ,Yt,r) and of the Perron eigenvector u= (u1, . . . ,ur) according to communication of types, we have Yt,i =Zt,iui, t∈R+, i∈ {1, . . . , r}, where (Zt,i)t∈R+, i∈ {1, . . . , r}, are independent squared Bessel processes. It is interesting to note that Kesten and Stigum [14] considered a supercritical indecomposable multi-type branching processes without immigration, and they proved that there exists a random variable w such that %(mξ)^{−(rn+i−1)}X_rn+i−1→wui almost surely as n→ ∞ for each i∈ {1, . . . , r}.

The results of the present paper will be very useful for analysing the asymptotic behavior of the conditional least squares estimators of parameters of a critical multi-type branching process with immigration when the process is indecomposable but not positively regular. The positively regular case has been studied in Isp´any et al. [7] and in K¨ormendi and Pap [15].

2. Indecomposable multi-type branching processes with immigration

Let R++ denote the set of positive real numbers. The d-dimensional unit matrix will be denoted by I_d. Every random variable will be defined on the fixed probability space (Ω,A,P).

In what follows we recall some known facts about irreducible nonnegative matrices. The matrix mξ is reducible if there exist a permutation matrix P ∈R^p×p and q∈ {1, . . . , p−1} such that

P^>mξP =

B C 0 D

,

where B ∈R^q×q, D ∈R(p−q)×(p−q), C ∈R^q×(p−q) and 0∈R^(p−q)×q is a null matrix. The matrix m_ξ is irreducible if it is not reducible; see, e.g., Horn and Johnson [6, Definition 6.2.21, Definition 6.2.22]. If the

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matrix m_ξ is irreducible then, by the Frobenius–Perron theorem the following assertions hold (see, e.g., Bapat and Raghavan [2, Theorem 1.8.3], Berman and Plemmons [2, Theorem 2.20], Brualdi and Cvetkovi´c [4, Theorem 8.2.4], Minc [16, Theorem 3.1] or Kesten and Stigum [14]) :

• %(m_ξ)∈R++, %(m_ξ) is an eigenvalue of m_ξ, and the algebraic and geometric multiplicities of %(m_ξ) equal 1.

• Corresponding to the eigenvalue %(m_ξ) there exists a unique (right) eigenvector u∈R^p++, called the Perron vector of m_ξ, such that the sum of the coordinates of u is 1, and there exists a unique left eigenvector v∈R^p++, such that u^>v = 1.

• If mξ has exactly r eigenvalues of maximum modulus %(mξ) then the coordinates {1, . . . , p} can be partitioned into r nonempty mutually disjoint subsets D1, . . . , Dr such that mi,j = 0 unless there exists `∈ {1, . . . , r} with i∈D`−1 and j ∈D`, where subscripts are considered modulo r.

This partitioning is unique up to cyclic permutation of the subsets. We may assume that the types are enumerated according to these subsets, and hence

mξ =







0 m1,2 0 · · · 0 0 0 0 m2,3 · · · 0 0

0 0 0 · · · 0 0

... ... ... . .. ... ... 0 0 0 · · · 0 mr−1,r

m_r,1 0 0 · · · 0 0







, (2.1)

where the r main diagonal zero blocks are square, m_1,2 ∈ R^|D¹^|×|D²^|, m_2,3 ∈ R^|D²^|×|D³^|, . . . , m_r,1∈R^|D^r^|×|D¹^| where |H| denotes the number of elements of a set H, and m_1,26=0, m_2,36=0, . . . , mr,16=0. Then for each k∈ {1, . . . , r−1}, we have

m^k_ξ=







0 · · · 0 fm1,k+1 0 · · · 0 0 · · · 0 0 mf2,k+2 · · · 0

... ... ... ... . .. ...

0 · · · 0 0 0 · · · fm_r−k,r

fmr−k+1,r+1 · · · 0 0 0 · · · 0

... . .. ... ... ... ...

0 · · · mfr,r+k 0 0 · · · 0







with

fmi,j :=mi,i+1mi+1,i+2· · ·mj−1,j ∈R^|Dⁱ^|×|D^j^|

for i, j∈N with i < j, where subscripts on the right hand side are considered modulo r. We will also use the notational convention fmi,i:=I_|D_i_|. Moreover,

m^r_ξ=







fm1,r+1 0 · · · 0 0 fm2,r+2 · · · 0 ... ... . .. ... 0 0 · · · fmr,2r







=:fm1,r+1⊕fm2,r+2⊕ · · · ⊕fmr,2r

The matrices fm_1,r+1 ∈R^|D¹^|×|D¹^|, fm_2,r+2∈R^|D²^|×|D²^|, . . . ,fm_r,2r∈R^|D^r^|×|D^r^| are primitive (that is, irreducible and there exists ni ∈N such that fmⁿ_i,r+iⁱ ∈R^|D++ⁱ^|×|Dⁱ^|) with %(fmi,r+i) = [%(mξ)]^r, i∈ {1, . . . , r}. (See, e.g., Minc [16, Theorem 4.3].)

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• If

u=





 u₁

... u_r





, v=





 v₁

... v_r







denotes the partitioning of u and v with respect to the partitioning D1, . . . , Dr of the coordinates {1, . . . , p} then, for each i∈ {1, . . . , r}, we have u^>_i vi =r⁻¹, the vectors uei:=rui and vei :=vi

are right and left eigenvectors of fm_i,r+i with ue^>_i ve_i= 1, and

[%(fmi,r+i)]⁻ⁿmfⁿ_i,r+i→ueiev^>_i =ruiv^>_i =:Πi∈R^|D++ⁱ^|×|Dⁱ^| as n→ ∞.

Consequently,

[%(mξ)]^−nrm^nr_ξ →Π∈R^p×p+ as n→ ∞, where

Π:=Π1⊕Π2⊕ · · · ⊕Πr=







Π1 0 · · · 0 0 Π2 · · · 0 ... ... . .. ... 0 0 · · · Πr







. (2.2)

The vectors u and v are right and left eigenvectors of Π corresponding to the eigenvalue [%(mξ)]^r.

• Moreover, there exist c, κ∈R++ with κ <1 such that

k[%(mξ)]^−nrm^nr_ξ −Πk6cκⁿ for all n∈N, (2.3) where kAk denotes the operator norm of a matrix A∈R^p×p defined by kAk:= sup_kxk=1kAxk.

If m_ξ has the form (2.1), then the offsprings have the property ξ_1,1,i,j= 0 almost surely unless there exists

`∈ {1, . . . , r} with i∈D_`−1 and j∈D_`, where D₀ :=D_r. Consequently, the offspring variance matrices V_ξ_j, j∈ {1, . . . , p}, have the form

Vξ_j =











0⊕0⊕ · · · ⊕0⊕V_1,j if j∈D₁, V2,j ⊕0⊕ · · · ⊕0⊕0 if j∈D2, ...

0⊕0⊕ · · · ⊕V_r,j⊕0 if j∈D_r,

(2.4)

where V_`,j ∈ R^|D^`−1^|×|D^`−1^| denotes the variance matrix of the random vector (ξ_1,1,j,i)_i∈D_`−1 for ` ∈ {1, . . . , r}, j ∈ D_`. For a vector α_` = (α_`,j)_j∈D_` ∈ R^|D+^`^| with ` ∈ {1, . . . , r}, we will use notation α_`V_` := P

j∈D`α_`,jV_`,j ∈ R^|D^`−1^|×|D^`−1^|, which is a positive semi-definite matrix, a mixture of the variance matrices V`,j, j ∈ D`. For a vector α = (αi)i∈{1,...,p} ∈ R^p+, we will also use the notation αVξ :=Pp

i=1αiVξ_i ∈R^p×p, which is a positive semi-definite matrix, a mixture of the variance matrices Vξ₁, . . . ,Vξ_p.

3. Convergence results

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 allR^d-valued càdlàg and continuous functions on R+, respectively. Let D_∞(R+,R^d) denote the Borelσ-algebra in D(R+,R^d) for the metric defined in Jacod and Shiryaev [10, Chapter VI, (1.26)] (with this metric D(R+,R^d) is a complete and separable metric space). ForR^d-valued stochastic processes (Y_t)_t∈_R₊ and (Y⁽ⁿ⁾_t )_t∈_R₊, n ∈ N, with càdlàg paths we write Y⁽ⁿ⁾ −→^D Y if the distribution

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of Y⁽ⁿ⁾ on the space (D(R+,R^d),D_∞(R+,R^d)) converges weakly to the distribution of Y on the space (D(R+,R^d),D_∞(R+,R^d)) as n→ ∞.

Theorem 3.1. Let (Xk)_k∈Z₊ be a indecomposable criticalp-type branching process with immigration. Suppose X₀ =0, E(kξ_1,1,ik²)<∞ for all i∈ {1, . . . , p} and E(kε1k²)<∞. Suppose that the index of cyclicity of mξ is r∈N. Suppose that the offspring mean matrix mξ has the form (2.1). For each n∈N, consider the rp-dimensional random step process

X⁽ⁿ⁾_t := 1 rn







X_rbntc X_rbntc−1

... X_rbntc−r+1







, t∈R+.

Then

X⁽ⁿ⁾−→^D X as n→ ∞, (3.1)

where

Xt:=





 m^r_ξYt

m^r−1_ξ Yt

... m_ξYt







, t∈R+, (3.2)

with

Yt:=





 Yt,1

Yt,2

... Yt,r







, t∈R+,

where, for i∈ {1, . . . , r}, the|Di|-dimensional process (Yt,i)_t∈R₊ is given by Yt,i:=Zt,iui, t∈R+,

where (Zt,i)_t∈_R₊ is the unique strong solution of the SDE dZt,i =v^>_i mξ,ε,idt+

q

v^>_i Vξ,ε,iviZ_t,i⁺dWt,i, t∈R+, (3.3) with initial value Z0,i= 0, where (Wt,i)_t∈_R₊, i∈ {1, . . . , r}, are independent standard Wiener processes and

mξ,ε,i:= 1 r

i+r−1

X

`=i

fmi,`mε,`, Vξ,ε,i:= 1 r

i+r−1

X

`=i

fmi,`[(mf`+1,i+rui)V`+1]fm^>_i,`, where

m_ε=





 m_ε,1 m_ε,2

... m_ε,r







denotes the partitioning of m_ε with respect to the partitioning D₁, . . . , D_r of the types {1, . . . , p}. Here the second subscript of m_ε,` in the definition of m_ξ,ε,i and the subscript of V_`+1 in the definition of V_ξ,ε,i are considered modulo r. Moreover, thep-dimensional coordinate blocks of therp-dimensional process (X_t)_t∈_R₊

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have the same distribution, i.e., (mⁱ_ξYt)_t∈_R₊ = (Y^D t)_t∈_R₊ for all i∈ {1, . . . , r−1}, and they are periodic, i.e., (m^r_ξYt)_t∈_R₊= (Yt)_t∈_R₊ almost surely.

Remark 3.2. The SDE (3.3) has a unique strong solution (Z_t,i^(z⁰⁾)_t∈R₊ for all initial values Z_0,i^(z⁰⁾=z0 ∈R. Indeed, the coefficient functions satisfy conditions of part (ii) of Theorem 3.5 in Chapter IX in Revuz and Yor [18]

or the conditions of Proposition 5.2.13 in Karatzas and Shreve [13]. Further, by the comparison theorem (see, e.g., Revuz and Yor [18, Theorem 3.7, Chapter IX]), if the initial value Z_0,i^(z⁰⁾=z0 is nonnegative, then Z_t,i^(z⁰⁾ is nonnegative for all t ∈R+ with probability one. Hence Z_t,i⁺ may be replaced by Z_t,i under the square

root in (3.3). 2

Remark 3.3. Note that Theorem 3.1 implies the convergence result of Isp´any and Pap [9] for a positively regular critical p-type branching process (X_k)_k∈_Z₊ with immigration. Indeed, in this case the index of cyclicity is r = 1, and mξ,ε,1 = mε,1, Vξ,ε,1 = uVξ. In fact, the result of Isp´any and Pap [9] has been proven under the higher moment assumptions E(kξ_1,1,ik⁴)<∞ for all i∈ {1, . . . , p} and E(kε1k⁴)<∞. Moreover, Theorem 3.1 also implies the convergence result of Barczy et al. [3, Theorem 3.1] for a primitive INAR(p) process. Eventually, Theorem 3.1 yields a convergence result for an arbitrary INAR(p) process as well. 2

In order to prove (3.1), introduce the rp-dimensional random vectors

M_k:=





 M_k,1 M_k,2

... Mk,r





 :=







X_rk−E(X_rk| F_rk−1) X_rk−1−E(X_rk−1| F_rk−2)

...

X_rk−r+1−E(X_rk−r+1| F_rk−r)







=







X_rk−m_ξX_rk−1−m_ε X_rk−1−m_ξX_rk−2−m_ε

...

X_rk−r+1−mξX_rk−r−mε







(3.4)

for k∈ N, forming a sequence of martingale differences with respect to the filtration (Frk)_k∈Z₊. Consider therp-dimensional random step processes

M⁽ⁿ⁾_t :=





 M⁽ⁿ⁾_t,1

... M⁽ⁿ⁾_t,r





 := 1

rn

bntc

X

k=1

Mk, t∈R+, n∈N.

The following convergence result is an important step in the proof of Theorem 3.1.

Theorem 3.4. Under the assumptions of Theorem 3.1, we have M⁽ⁿ⁾−→^D M as n→ ∞, where

Mt:=





 Mt,1

... M_t,r





 t∈R+, is the unique strong solution of the SDE

dMt,i =1 r

v u u u t



m^r−i_ξ Π

r

X

j=1

m^j−1_ξ (rMt,j+tm_ε)





+

V_ξdWt,i, i∈ {1, . . . , r}, (3.5)

with initial value M0 = 0, where (Wt,i)_t∈R₊, i ∈ {1, . . . , r}, are independent p-dimensional standard Wiener processes, and for a positive semi-definite matrix A ∈ R^p×p, √

A denotes its unique symmetric positive semi-definite square root.

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In order to handle the SDE (3.5), consider thep-dimensional process

N_t:=M_t,1+m_ξM_t,2+· · ·+m^r−1_ξ M_t,r, t∈R+. (3.6) Theorem 3.5. Under the assumptions of Theorem 3.1, the process (Nt)_t∈R₊ is the unique strong solution of the SDE

dN_t= 1 r

r

X

j=1

m^j−1_ξ v u u t

m^r−j_ξ Π

rN_t+t

r

X

`=1

m^`−1_ξ m_ε ⁺

V_ξdW_t,j, t∈R+, (3.7) with initial value N₀=0, and

Mt,i=1 r

Z t 0

v u u t

m^r−i_ξ Π

rNs+s

r

X

`=1

m^`−1_ξ mε

+

VξdWs,i, i∈ {1, . . . , r}. (3.8)

If

N_t=





 N_t,1

... Nt,r





, W_t,j =





 W_t,j,1

... Wt,j,r





, j∈ {1, . . . , r}, t∈R+,

denote the partitioning of Nt and Wt,j, j∈ {1, . . . , r}, with respect to the partitioning D1, . . . ,Dr of the coordinates {1, . . . , p}, then the process (Nt)_t∈_R₊ is the unique strong solution of the SDE

dN_t,i= v u u t

v^>_i

N_t,i+ t r

i+r−1

X

`=i

fm_i,`m_ε,`

⁺ ^i+r−1

X

`=i

mf_i,`

q

(fm_`+1,iu_i)V_`+1dWt,`+i−1,`+1, (3.9)

i∈ {1, . . . , r}, where the second subscript of m_ε,`, and the second and third subscripts of Wt,`+i−1,`+1 are considered modulo r.

From (3.4) we obtain the recursion X_rk−i+1=mξX_rk−i+Mk,i+mε for k∈N and i∈ {1, . . . , r}, and hence

X_r−i+1 =

r

X

`=i

m^`−i_ξ (M_1,`+m_ε) for i∈ {1, . . . , r}, and

X_rk−i+1=m^r_ξX_rk−r−i+1+

r

X

`=i

m^`−i_ξ (Mk,`+mε) +

i−1

X

`=1

m^`−i+r_ξ (M_k−1,`+mε), for k∈N with k>2 and i∈ {1, . . . , r}. This recursion implies

X_rk−i+1=

k

X

j=1

m^(k−j)r_ξ

r

X

`=i

m^`−i_ξ (Mj,`+mε) +

k

X

j=2

m^(k−j)r_ξ

i−1

X

`=1

m^`−i+r_ξ (M_j−1,`+mε) (3.10)

for k∈N and i∈ {1, . . . , r}. Applying Lemma 7.4, which is a version of the continuous mapping theorem, together with (3.10), (3.6) and Theorem 3.4, we show the following convergence result.

Theorem 3.6. Under the assumptions of Theorem 3.1, we have X⁽ⁿ⁾−→^D X as n→ ∞,

(9)

where X = (X_t)_t∈_R₊ is given by (3.2)with Y_t:=Π

N_t+ t

r

X

`=1

m^`−1_ξ m_ε

, t∈R+,

for which we have (m^r_ξY_t)_t∈_R₊= (Y_t)_t∈_R₊ almost surely.

Proof of Theorem 3.1. Theorem 3.1 is an easy consequence of Theorems 3.5 and 3.6. Indeed, Π=Π₁⊕· · ·⊕Π_r and Π_i =ru_iv^>_i , rv^>_i u_i = 1 for all i∈ {1, . . . , r}, hence we conclude from Theorems 3.5 and 3.6 that for each i∈ {1, . . . , r}, the process Zt,i:=v^>_i Yt,i, t∈R+, satisfies

Zt,i=v^>_i Πi

Nt,i+ t r

i+r−1

X

`=i

mfi,`mε,`

=v^>_i

Nt,i+t r

i+r−1

X

`=i

fmi,`mε,`

, t∈R⁺,

hence

Zt,iui=uiv^>_i

Nt,i+ t r

i+r−1

X

`=i

fmi,`mε,`

=Πi

Nt,i+ t r

i+r−1

X

`=i

mfi,`mε,`

=Yt,i. By Itˆo’s formula, we obtain that (Zt,i)_t∈_R₊ is a strong solution of the SDE

dZt,i =r⁻¹v^>_i

i+r−1

X

`=i

fm_i,`m_ε,`dt+v^>_i q r⁻¹Z_t,i⁺

i+r−1

X

`=i

fm_i,`

q

(fm_`+1,iu_i)V_`+1dWt,`+i−1,`+1 (3.11) with initial value Z0,i= 0. This equation can be written in the form (3.3), where (Wt,i)_t∈_R₊, i∈ {1, . . . , r}, are independent standard Wiener processes. Indeed, we have

rv^>_i Vξ,ε,ivi=v^>_i

i+r−1

X

`=i

mfi,`[(fm`+1,iui)V`+1]fm^>_i,`vi

=

i+r−1

X

`=i

v^>_i mf_i,`

q

(fm_`+1,iu_i)V_`+1

v^>_i mf_i,`

q

(fm_`+1,iu_i)V_`+1 >

=

i+r−1

X

`=i

v^>_i fmi,`

q

(fm`+1,iui)V`+1

2

.

Hence, if v^>_i fm_i,`p

(fm_`+1,iu_i)V_`+1=0 for each `∈ {i, . . . , i+r−1}, then (3.3) trivially follows, and if there exists `∈ {i, . . . , i+r−1} with v^>_i fm_i,`p

(fm_`+1,iu_i)V_`+16=0, then (3.3) holds with Wt,i:=v^>_i Pi+r−1

`=i fmi,`

p(mf`+1,iui)V`+1Wt,`+i−1,`+1

v^>_i Pi+r−1

`=i fmi,`[(fm`+1,iui)V`+1]fm^>_i,`vi

, t∈R+, i∈ {1, . . . , r},

which are independent standard Wiener processes, since

(`+i−1, `+ 1) :`∈ {i, . . . , i+r−1} , i∈ {1, . . . , r},

are disjoint sets. Consequently, we conclude (3.1). 2

Remark 3.7. An alternative way of proving Theorem 3.1 is first checking that

Yk:=





 Xrk

X_rk−1 ... X_rk−r+1







, k∈Z⁺,

(10)

is a positively regular critical rp-type branching process with immigration (it can be done, for instance, by the help of generating functions), then determining the immigration mean vector and the offspring variation matrices of (Y_k)_k∈_Z₊, and then applying the convergence result of Isp´any and Pap [9]. This would have been

also a cumbersome calculation. 2

4. Proof of Theorem 3.5

If (Mt)_t∈R₊ is a strong solution of the SDE (3.5), then the process (Nt)_t∈R₊ is a strong solution of the SDE (3.7) with initial value N0=0, and (3.8) trivially holds.

Using the block matrix form of mξ, Π and Vξ₁, . . . ,Vξ_p (see (2.1), (2.2) and (2.4)), we obtain

dN_t,i=1 r

r

X

j=1

fm_i,i+j−1 v u u t

fm_i−r+j,iΠ_i

rN_t,i+t

i+r−1

X

`=i

fm_i,`m_ε,`

⁺

V_i−r+jdW_t,j,i+j (4.1)

for each i∈ {1, . . . , r}. Indeed, the covariance matrices Vξ_j j ∈ {1, . . . , p}, have block-diagonal form, see (2.4), hence

v u u t

m^r−j_ξ Π

rNt+t

r

X

`=1

m^`−1_ξ m_ε +

V_ξ

=

2r−j+1

M

i=r−j+2

v u u t

fm_i−r+j,iΠi

rNt,i+t

r+i−1

X

`=i

mfi,`mε,`

⁺

V_i−r+j,

where we also used that for an arbitrary matrix A∈R^p×p with partitioning

A=







A1,1 · · · A1,r

... . .. ... Ar,1 · · · Ar,r







with respect to the partitioning D1, . . . ,Dr of the coordinates {1, . . . , p}, we have

m^k_ξA=







fm1,k+1Ak+1,1 · · · mf1,k+1Ak+1,r

... . .. ... fm_r,k+rA_k+r,1 · · · fm_r,k+rA_k+r,r







for all k∈ {1, . . . r−1}, where the first subscript of A_i,j is considered modulo r. Substituting this into (3.7) and using again the above block form of m^k_ξA for A∈R^p×p and k∈ {1, . . . r−1}, we obtain (4.1).

Using Π_i=ru_iv^>_i for all i∈ {1, . . . , r}, equation (4.1) can be written in the form (3.9). 2

5. Proof of Theorem 3.4

In order to prove M⁽ⁿ⁾ −→^D M, we want to apply Theorem 7.3 for U = M, U⁽ⁿ⁾_k = n⁻¹Mk and F_k⁽ⁿ⁾:=F_k for n∈N and k∈Z+, and with coefficient function γ:R+×(R^p)^r→(R^p×p)^r×r of the SDE

(11)

(3.5) given by

γ(t,x) =1 r

r

M

i=1

v u u t

m^r−i_ξ Π

r

X

j=1

m^j−1_ξ (rxj+tmε) ⁺

Vξ, x=





 x1

... xr





∈(R^p)^r.

The aim of the following discussion is to show that the SDE (3.5) has a unique strong solution M^(x_t ⁰⁾

t∈R+

with initial value M^(x₀ ⁰⁾ = x0 for all x0 ∈ (R^p)^r. Clearly, it is sufficient to prove that the SDE (3.7) has a unique strong solution (N^(y_t ⁰⁾)_t∈_R₊ with initial value N^(y₀ ⁰⁾ = y₀ for all y₀ ∈ (R^p)^r. Indeed, if

M^(x_t ⁰⁾

t∈R+ is a strong solution of the SDE (3.5) with initial value M^(x₀ ⁰⁾=x₀ with some

x₀=





 x_0,1

... x0,r





∈(R^p)^r,

then Nt := Pr

i=1mⁱ⁻¹_ξ M^(x_t,i⁰⁾ is a strong solution of the SDE (3.7) with initial value Pr

i=1mⁱ⁻¹_ξ x_0,i. Conversely, if N^(y_t ⁰⁾

t∈R+ is a strong solution of the SDE (3.7) with initial value N^(y₀ ⁰⁾=y₀ with some y₀∈R^p then with

x0=





 x0,1

x0,2

... x_0,r





 :=





 y₀

0 ... 0





 ,

we have y₀=Pr

i=1mⁱ⁻¹_ξ x0,i∈R^p, and M_t,i:=x_0,i+1

r Z t

0

v u u t

m^r−i_ξ Π

rN^(y_s ⁰⁾+s

r

X

`=1

m^`−1_ξ m_ε ⁺

V_ξdW_s,i, i∈ {1, . . . , r},

is a strong solution of the SDE (3.5) with initial value x0.

Hence it is enough to show that the SDE (3.7) has a unique strong solution N^(y_t ⁰⁾

t∈R+ with initial value N^(y₀ ⁰⁾=y₀ for all y₀∈R^p. First observe that if N^(y_t,i^0,i⁾

t∈R+ is a strong solution of the SDE (3.9) with initial value N^(y_0,i^0,i⁾=y_0,i∈R^|Dⁱ^|, then, by Itˆo’s formula, the process (Pt,i,Qt,i)_t∈R₊, defined by

Pt,i:=v^>_i

N^(y_t,i^0,i⁾+t r

i+r−1

X

`=i

fmi,`mε,`

, Qt,i :=N^(y_t,i^0,i⁾− Pt,iui

is a strong solution of the SDE











dP_t,i = ¹_rv^>_i Pi+r−1

`=i fm_i,`m_ε,`dt +q

r⁻¹P_t,i⁺ v^>_i Pi+r−1

`=i fmi,`

p(fm`+1,i+rui)V`+1dWt,`+i−1,`+1, dQt,i =−¹_rΠiPi+r−1

`=i fmi,`mε,`dt +q

r⁻¹P_t,i⁺ (I_|D_i_|−Πi)Pi+r−1

`=i mfi,`

p(fm`+1,i+rui)V`+1dWt,`+i−1,`+1

(5.1)

(12)

with initial value (P_0,i,Q_0,i) = (v^>_i y_0,i,(I_|D_i_|−Π_i)y_0,i). Indeed, the first SDE of (5.1) is an easy consequence of the SDE (3.9). The second one can be checked as follows. By Itˆo’s formula,

dQ_t,i= dN^(y_t,i^0,i⁾−u_idP_t,i= dN^(y_t,i^0,i⁾−u_iv^>_i

dN^(y_t,i^0,i⁾+1 r

i+r−1

X

`=i

fm_i,`m_ε,`dt

=−1 rΠi

i+r−1

X

`=i

mfi,`mε,`dt+ (I_|D_i_|−Πi) dN^(y_t,i^0,i⁾

with Q0,i = N^(y_0,i^0,i⁾− P0,iui = y_0,i−(v^>_i y_0,i)ui = y_0,i−uiv^>_i y_0,i = (I_|D_i_|−Πi)y_0,i. Conversely, if (P_t,i^(p^0,i^,q^0,i⁾,Q^(p_t,i^0,i^,q^0,i⁾)_t∈R₊ is a strong solution of the SDE (5.1) with initial value P_0,i^(p^0,i^,q^0,i⁾,Q^(p_0,i^0,i^,q^0,i⁾

= (p0,i,q_0,i)∈R×R^|Dⁱ^|, then, again by Itˆo’s formula,

Nt,i:=P_t,i^(p^0,i^,q^0,i⁾ui+Q^(p_t,i^0,i^,q^0,i⁾, t∈R+,

is a strong solution of the SDE (3.9) with initial value N0,i =p0,iui+q_0,i. The correspondence y_0,i ↔ (p0,i,q_0,i) := (v^>_i y_0,i,(I_|D_i_|−Πi)y_0,i) is a bijection between R^|Dⁱ^| and R× {q∈R^|Dⁱ^|:v^>_i q = 0}, since y_0,i=p0,iui+q_0,i, and

(I_|D_i_|−Πi)(p0,iui+q_0,i) =p0,iui+q_0,i−Πip0,iui+Πiq_0,i=p0,iui+q_0,i−p0,iuiv^>_i ui+uiv^>_i q_0,i=q_0,i. The right hand side of the SDE (5.1) contains only the process (P_t,i)_t∈_R₊, hence it is enough to show that the first equation of (5.1) has a unique strong solution (P_t,i^(p^0,i^,q^0,i⁾)_t∈R₊ with initial value P_0,i^(p^0,i^,q^0,i⁾ =p0,i

for all p0,i∈R. The first equation of (5.1) is the same as (3.11), which can be written in the form (3.3), see the end of Section 3. Hence, by Remark 3.2, the first equation of the SDE (5.1) has a unique strong solution (P_t,i^(p^0,i⁾)_t∈R₊ with initial value P_0,i^(p^0,i⁾=p0,i for all p0,i ∈R. Consequently, the SDE (5.1), and hence the SDE (3.5) admit a unique strong solution with arbitrary initial value.

Now we show that conditions (i) and (ii) of Theorem 7.3 hold. We have to check that for each T >0,

sup

t∈[0,T]

1 (rn)²

bntc

X

k=1

E

M_kM^>_k Frk−r

− Z t

0

(R⁽ⁿ⁾_s )⁺ds

−→P 0, (5.2)

1 (rn)²

bnTc

X

k=1

E kM_kk²1_{kM_k_k>nθ}

F_rk−r−→P 0, for all θ >0, (5.3)

as n→ ∞, where the process (R⁽ⁿ⁾_s )_s∈R₊ is defined by

R⁽ⁿ⁾_s := 1 r²

r

M

i=1

m^r−i_ξ Π

r

X

j=1

m^j−1_ξ (M⁽ⁿ⁾_s,j +r⁻¹smε)

Vξ

(13)

for s∈R+, n∈N. By (3.4), Π

r

X

j=1

m^j−1_ξ (M⁽ⁿ⁾_s,j +r⁻¹smε)

=Π

r

X

j=1

m^j−1_ξ

(rn)⁻¹

bnsc

X

k=1

(X_rk−j+1−mξX_rk−j−mε) +r⁻¹smε

= (rn)⁻¹Π

bnsc

X

k=1 r

X

j=1

(m^j−1_ξ Xrk−j+1−m^j_ξXrk−j−m^j−1_ξ mε) +r⁻¹sΠ

r

X

j=1

m^j−1_ξ mε

= (rn)⁻¹Π

bnsc

X

k=1

Xrk−m^r_ξX_rk−r−

r

X

j=1

m^j−1_ξ mε

+r⁻¹sΠ

r

X

j=1

m^j−1_ξ mε

= (rn)⁻¹

bnsc

X

k=1

ΠX_rk−ΠX_rk−r−Π

r

X

j=1

m^j−1_ξ m_ε

+r⁻¹sΠ

r

X

j=1

m^j−1_ξ m_ε

= (rn)⁻¹ΠX_rbnsc+

s−bnsc n

r⁻¹Π

r

X

j=1

m^j−1_ξ m_ε,

where we used

Πm^r_ξ=

n→∞lim m^nr_ξ

m^r_ξ= lim

n→∞m^(n+1)r_ξ =Π. (5.4)

Consequently,

R⁽ⁿ⁾_s = 1 r²

r

M

i=1

n⁻¹m^r−i_ξ ΠX_rbnsc+

s−bnsc n

Π

r

X

j=1

m^j−1_ξ m_ε

V_ξ

,

since

m^r−i_ξ Π=m^r−i_ξ

n→∞lim m^nr_ξ

=

n→∞lim m^nr+r−i_ξ

=

n→∞lim m^nr_ξ

m^r−i_ξ =Πm^r−i_ξ and (5.4) implies

m^r−i_ξ Π

r

X

j=1

m^j−1_ξ =Πm^r−i_ξ

r

X

j=1

m^j−1_ξ =Π ⁱ

X

j=1

m^j−1+r−i_ξ +m^r_ξ

r

X

j=i+1

m^j−1−i_ξ

=Π

r

X

j=1

m^j−1_ξ .

Thus (R⁽ⁿ⁾_t )⁺=R⁽ⁿ⁾_t , and Z t

0

(R⁽ⁿ⁾_s )⁺ds=

r

M

i=1

1

(nr)²m^r−i_ξ Π

bntc−1

X

`=0

Xr`+nt− bntc

(nr)² m^r−i_ξ ΠX_rbntc +bntc+ (nt− bntc)²

2(nr)² Π

r

X

j=1

m^j−1_ξ mε

Vξ

.

Using (7.3), we obtain 1

(nr)²

bntc

X

k=1

E(MkM^>_k | F_rk−r) =

r

M

i=1

bntc

n² Vε+ 1 n²

bntc

X

k=1

m^r−i_ξ X_rk−r+

r−i

X

j=1

m^j−1_ξ mε Vξ

.