A limit theorem of two-type Galton–Watson branching processes with immigration

Statistics and Probability Letters 79 (2009) 1710–1716

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A limit theorem of two-type Galton–Watson branching processes with immigration Chunhua Ma School of Mathematical Sciences and LPMC, Nankai University, Tianjin 300071, PR China

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Article history: Received 15 March 2009 Received in revised form 11 April 2009 Accepted 13 April 2009 Available online 3 May 2009 MSC: primary 60J35 secondary 60J80 60H20 60K37

abstract We provide a simple set of sufficient conditions for the weak convergence of two-type Galton–Watson branching processes with immigration to two-dimensional, continuoustime, continuous-state branching processes with immigration, which generalizes the limit result of Li [Li, Z.H., 2006a. A limit theorem of discrete Galton–Watson branching processes with immigration. J. Appl. Probab. 43, 289– 295]. © 2009 Elsevier B.V. All rights reserved.

1. Introduction A continuous-state branching process with immigration (CBI-process) was first introduced by Kawazu and Watanabe (1971). They showed that the class of Markov processes may arise as the high density limits in finite-dimensional distributions of a sequence of Galton–Watson processes with immigration (GWI-processes). Such results have become the basis of many studies of CBI-processes; see e.g. Pitman and Yor (1982) and Shiga and Watanabe (1973). However, the conditions for the above convergence in Kawazu and Watanabe (1971) involve iterations of the probability generating functions, and this is sometimes not easy to verify. Hence some simpler conditions were recently provided in Li (2006a), which further ensure that the weak convergence of GWI-processes also holds on the space of càdlàg path D([0, ∞), R+ ). The main objective of this note is to extend Li’s result to a multi-type process. For simplicity, we only consider the case of two dimensions though many arguments can be carried over to the case of higher dimensions. Actually, we provide a set of relatively simple sufficient conditions for the weak convergence of the two-type GWI-processes (re-scaled in time and space) to the two-dimensional CBI-processes. In the two-type case, we face nontrivial difficulties which mainly arise from some kinds of representation problems (see Remark 2.1 for details) and thus the crucial part of our proof, partly inspired by Venttsel’ (1959), consists in obtaining a Lévy–Khintchine-type representation of the limit of the re-scaled bivariate probability generating functions. As a corollary of this result, we also show that one of the hypotheses can be removed in Theorem 2.1 of Li (2006a). The concept of two-dimensional CB-processes was introduced by Watanabe (1969) and the corresponding processes with immigration have been systematically studied by several authors in the measure-valued setting; see, e.g., Li (2006b) for the survey on the topic. Two-dimensional CBI-processes are included in a wide class of so-called affine processes and many interesting applications of the class of processes have been found in mathematical finance; see Duffie et al. (2003). In this note we will consider a special class of two-dimensional CBI-processes. Before continuing with our introduction we need to introduce the following notations and definitions. For u ∈ R set l1 (u) = |u|, l12 (u) = |u| ∧ |u|2 and χ (u) = (1 ∧ u) ∨ (−1).

E-mail address: [email protected]. 0167-7152/$ – see front matter © 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.spl.2009.04.008

C. Ma / Statistics and Probability Letters 79 (2009) 1710–1716

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For u = (u1 , u2 ) ∈ R2 define χ (u) = (χ (u1 ), χ (u2 )). Suppose that R(1) and R(2) are functions on R2+ defined by R

(1)

(λ1 , λ2 ) = a11 λ1 + a12 λ2 − αλ − 2 1

Z

e−hλ,ui − 1 + λ1 u1 µ1 (du),

(1.1)

e−hλ,ui − 1 + λ2 u2 µ2 (du),

(1.2)




R2

+

R(2) (λ1 , λ2 ) = a21 λ1 + a22 λ2 − βλ22 −

Z




R2+

where (aij ) is a (2 × 2)-matrix with a12 ≥ 0 and a21 ≥ 0, real constants α, β ≥ 0, and µ1 , µ2 are σ -finite measures on R2+ supported by R2+ \ {0} such that

Z

Z

l12 (u1 ) + l1 (u2 ) µ1 (du) +




R2+

l1 (u1 ) + l12 (u2 ) µ2 (du) < ∞.




R2+

Furthermore, suppose that F is a function on R2+ defined by F (λ1 , λ2 ) = hb, λi +

Z R2+

(1 − e−hλ,ui )m(du),

(1.3)

where b = (b1 , b2 ) ∈ R2+ , and m(du) is a σ -finite measure on R2+ supported by R2+ \ {0} such that

Z R2+

  (1 ∧ u1 ) + (1 ∧ u2 ) m(du) < ∞.

A Markov process {Y (t ) = (Y1 (t ), Y2 (t )) : t ≥ 0} on R2+ is called a two-dimensional CBI-process if it has transition semigroup (Pt )t ≥0 given by

(

Z e

−hλ,ui

R2+

Pt (x, dy) = exp −hx, Ψ (t , λ)i −

)

t

Z

F (Ψ (s, λ))ds ,

λ ∈ R2+ ,

(1.4)

0

where Ψ (t , λ) = ψ1 (t , λ), ψ2 (t , λ) satisfies the backward equations




dψ1

dψ2 (t , λ) = R(1) (Ψ (t , λ)), (t , λ) = R(2) (Ψ (t , λ)), Ψ (0, λ) = λ. (1.5) dt dt In particular, if F (λ) ≡ 0, we simply call {Y (t ) : t ≥ 0} a two-dimensional CB-process. In addition, by Proposition 9.1 in Duffie et al. (2003), it is easy to see that the above CBI-process Y (·) is conservative, that is, Pt (x, ∆) = 0, for (t , x) ∈ R+ × R2+ , where R2+ ∪ {∆} is the one-point compactification of R2+ . The remainder of this note is organized as follows. In Section 2, we provide a very brief introduction to two-type GWI-processes and the main limit theorem (Theorem 2.1). In Section 3, we use a modified Venttsel’ method to specify representations of a class of continuous functions. Those functions give a complete characterization of the branching mechanisms for two-dimensional CBI-processes and arise as the limits of sequences involving probability generating functions. Based on those results, we complete the proof of the main result in Section 4. 2. Two-type GWI-processes and limit theorems Let N = {0, 1, 2, . . .}. Denote by N2 the set of all the 2 tuples i = (i1 , i2 ) where each element iη ∈ N(η = 1, 2). Let e(1) = (1, 0) and e(2) = (0, 1). By a two-type GWI-process we mean a discrete-time Markov chain {Y (n) = (y1 (n), y2 (n)) : n ∈ N} defined on N2 consisting of a two-type Galton–Watson branching process {Z (n) = (z1 (n), z2 (n)) : n ∈ N} augmented by an independent random immigration component at each generation. For the Galton–Watson branching process Z (n), we are given an offspring vector of bivariate probability generating functions G(s1 , s2 ) = g (1) (s1 , s2 ), g (2) (s1 , s2 )




with g (l) (s1 , s2 ) =

X

p(l) (i)s11 s22 , i

i

0 ≤ s1 , s2 ≤ 1 ,

i∈N2

where p(l) (i) = P Y (1) = i | Y (0) = e(l) , i ∈ N2 , l = 1, 2. For the immigration component, we are given the bivariate p.g.f. h(s1 , s2 ). Then the one-step transition matrix P (i, j) of the GWI-process {Y (n)} corresponding to the parameters (G, h) is given by




X

P (i, j)s11 s22 = g (1) (s1 , s2 ) j

j


i1


i2

g (2) (s1 , s2 )

h(s1 , s2 ),

i ∈ N2 .

j∈N2

Now let us consider a sequence of two-type GWI-processes {Yk (n) : n ∈ N} corresponding to the parameters (Gk , hk ). Let {γk } be a sequence of positive numbers. Set

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C. Ma / Statistics and Probability Letters 79 (2009) 1710–1716

"

(l)

Rk (λ1 , λ2 ) = kγk



1−

λl  k

(l)

− gk

1−

λ1 k

,1 −

λ2

!# ,

k

l = 1, 2,

(2.1)

and

" Fk (λ1 , λ2 ) = γk 1 − hk 1 −

λ1 k

,1 −

λ2 k

!# ,

(2.2)

for 0 ≤ λ1 , λ2 ≤ k. Let us consider the following conditions: (A) as k → ∞, we have γk → ∞; (l) (B) for l = 1, 2, the sequence {Rk } is uniformly Lipschitz in (λ1 , λ2 ) on each bounded rectangle, and converges to a continuous function as k → ∞; (C) the sequence {Fk } converges to a continuous function as k → ∞. (1)

(2)

Proposition 2.1. (i) Under condition (B), the limit functions R(1) and R(2) of Rk and Rk have representations (1.1) and (1.2) respectively.  (ii) For any functions R(1) and R(2) given by (1.1) and (1.2), there are sequences {γk } and Gk = (gk(1) , gk(2) ) as above such



(1)

that (A) and (B) hold with Rk



→ R(1) and R(k2) → R(2) .

Proposition 2.2. (i) Under condition (C), the limit function F of {Fk } has representation (1.3). (ii) For any function F given by (1.3), there are sequences {γk } and {hk } as above such that (A) and (C) hold with Fk → F . The main result of this paper is the following. Theorem 2.1. (i) Assume that (A), (B) and (C) hold. If Yk (0)/k converges in distribution to Y (0), then {Yk ([γk t ])/k : t ≥ 0} converges in distribution on D([0, ∞), R2+ ) to the two-dimensional CBI-process {Y (t ) : t ≥ 0} corresponding to ((R(1) , R(2) ), F ) with initial value Y (0). (ii) If {Y (t ) : t ≥ 0} is any two-dimensional CBI-process corresponding to ((R(1) , R(2) ), F ), there exist a sequence of positive numbers {γk } and a sequence of two-type GWI-processes {Yk (n) : n ∈ N} such that the sequence {Yk ([γk t ])/k : t ≥ 0} converges in distribution on D([0, ∞), R2+ ) to the process {Y (t ) : t ≥ 0}. The above theorem reveals a connection between two-type GWI-processes and two-dimensional CBI-processes and then (1) (1) (1) generalizes the limit result in Li (2006a). Indeed, Let gˆk (s1 ) = gk (s1 , 1), hˆ k (s1 ) = hk (s1 , 1), Rˆ k (λ1 ) = Rk (λ1 , 0) and (1)

Fˆk (λ1 ) = Fk (λ1 , 0). Note that gˆk and hˆ k are the marginal probability generating functions of gk

(1)

and hk , respectively.

Then we have a sequence of single-type GWI-processes {yk (n) : n ∈ N} corresponding to (ˆgk , hˆ k ). It is easy to check that Rˆ k (λ1 ) = kγk [(1 − λ1 /k) − gˆk (1 − λ1 /k)] and Fˆk (λ1 ) = γk [1 − hˆ k (1 − λ1 /k)]. In this case, conditions (B) and (C) as above can be rewritten in the following forms: ˆ the sequence {Rˆ k } is uniformly Lipschitz on each bounded interval and converges to a continuous function as k → ∞; (B) ˆ the sequence {Fˆk } converges to a continuous function as k → ∞. (C) ˆ and (C), ˆ the limit functions Rˆ and Fˆ of {Rˆ k } and {Fˆk } have the By Propositions 2.1 and 2.2, we have that under conditions (B) representations which coincide with R(λ1 , 0) and F (λ1 , 0) respectively. Thus by Theorem 2.1 as above, we have the following corollary. It removes one of the hypotheses which assumes that γk /k → γ0 for some γ0 ≥ 0 in Theorem 2.1 of Li (2006a).

Corollary 2.1. Suppose that (A), (Bˆ ) and (Cˆ ) hold. If yk (0)/k converges in distribution to y(0), then {yk ([γk t ])/k : t ≥ 0} converges in distribution on D([0, ∞), R+ ) to the one-dimensional CBI-process {y(t ) : t ≥ 0} corresponding to (Rˆ , Fˆ ) with initial value y(0). Remark 2.1. The ‘simplicity’ of conditions given in Corollary 2.1 lies in the representations of a class of continuous functions arising as the limits of sequences involving re-scaled probability generating functions. Li (1991) used a simple method based on Bernstein polynomials to obtain the results. However, the method of Li (1991) cannot apply to the two-dimensional case,  (1) i.e., Proposition 2.1. Actually, by the above method, we have that the limit function R(1) of Rk has the form (1)

R

(λ1 , λ2 ) = a11 λ1 + a12 λ2 − αλ − αλ ˆ 1 λ2 − 2 1

Z

e−hλ,ui − 1 + λ1 u1 µ1 (du),




R2

+

where αˆ ≥ 0, and a11 , a12 , α and µ1 are given as in (1.1). But in this way we cannot ensure that αˆ = 0. Inspired by Venttsel’ (1959), the proposition is proved in the next section. 3. Representation results for limit functions In this section, we introduce and prove some representation results for the limit functions of (2.1) and (2.2), which will play an important role in the proof of Theorem 2.1.

C. Ma / Statistics and Probability Letters 79 (2009) 1710–1716

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Proof of Proposition 2.1. We only prove the representation result for R(1) . The representation for R(2) can be proved in a (1) (1) (1) similar way. Let νk be the probability measure on N2 corresponding to gk . For simplicity, we write gk = gk , Rk = Rk and (1) R=R .   (i) Set Φk (λ1 , λ2 ) = kγk e−λ1 /k − gk (e−λ1 /k , e−λ2 /k ) . By the mean-value theorem, we have that Rk (λ1 , λ2 ) = Φk (λ1 , λ2 ) + kγk 1 − gk0 ,1 (ηk,1 , ηk,2 ) (1 − λ1 /k − e−λ1 /k )





− kγk gk0 ,2 (ηk,1 , ηk,2 )(1 − λ2 /k − e−λ2 /k ),

(3.1)

where 1 − λi /k ≤ ηk,i ≤ e−λi /k and gk0 ,i denotes the partial derivative of gk with respect to λi for i = 1, 2. Under condition (B), the sequences

|R0k,1 (λ1 , λ2 )| = γk 1 − gk0 ,1 (1 − λ1 /k, 1 − λ2 /k) , |R0 (λ1 , λ2 )| = γk g 0 (1 − λ1 /k, 1 − λ2 /k) k,2

k,2

k(1−η

)

k,i and k(1 − eλi /k ) ≤ k(1 − ηk,i ) ≤ k 0 0 λi , we have that the sequences γk 1 − gk,1 (ηk,1 , ηk,2 ) and γk gk,2 (ηk,1 , ηk,2 ) are also uniformly bounded. By (B) and (3.1), we have Φk (λ1 , λ2 ) −→ R(λ1 , λ2 ), as n → ∞. Thus it is enough to consider the limit representation of Φk . The following arguments are partly inspired by Venttsel’ (1959) (see also Li and Ma (2008)):
Step 1: Fix k ≥ 1. Let G = [−1, ∞) × [0, ∞) and Gk = { (i − 1)/k, j/k : i, j ∈ N}. Let ρk be the measure defined by

are uniformly bounded on each bounded rectangle [0, c ]2 for c ≥ 0. Since ηk,i = 1 −

ρk (·) = kγk

∞ X i ,j = 0

νk ({(i, j)})δ i−1 , j  (·). k

(3.2)

k

Then ρk is a finite measure on G supported by Gk . Let l(u) = (u21 + |u2 |) ∧ 1. Set %k = G l(u)ρk (du). If %k > 0, define Pk (du) = (l(u)/%k ) ρk (du). If %k = 0, we let Pk (·) be the Dirac measure at some point u0 ∈ Gk \ {0}. In both cases, we have that Pk (·) is a probability measure on G supported by Gk \ {0} and then

R

eλ1 /k Φk (λ1 , λ2 ) = βk,1 λ1 − %k

Z


−1

h(u, λ) l(u)

Pk (du).

(3.3)

G\{0}

where βk,1 =

χ (u1 )ρk (du) and h(u, λ) = e−hλ,ui − 1 + χ (u1 )λ1 . Step 2: Let G = G ∪{∆} be the one-point compactification of G. Then {Pk } is the sequence of probability measures on G∆ ; so it is relatively compact. Choose any subsequence denoted again by {Pk }, which converges to a probability P on G∆ .
Since each Pk is supported by Gk , we have that P is supported by R2+ ∪ {∆}. Let E be the set of ε > 0 for which P kuk = ε = 0. For ε ∈ E, we define a compact space of G∆ by Q := {u ∈ G : kuk ≤ ε} and Q is a P-continuity set. We have R

G

∆

(1)

(2)

eλ1 /k Φk (λ1 , λ2 ) = βk,1 λ1 + %k ak, ε λ2 − %k ak, ε λ21 + Ik, ε + Jk, ε ),

(3.4)

where (1) ak, ε

=

1 2

Z

−1

Q \{0}



Z Ik, ε =

h(u, λ) + χ (u2 )λ2 −

Q \{0}

Z Jk, ε =

(2) ak, ε

χ (u1 )(l(u)) Pk (du), 2

G\ Q

1 2

χ (u1 )λ 2

2 1

Z

χ (u2 )(l(u))−1 Pk (du),

= Q \{0}



(l(u))−1 Pk (du),

h(u, λ)(l(u))−1 Pk (du).

Furthermore we get limk→∞ Ik, ε = Q h(u, λ) + χ (u2 )λ2 − 21 χ 2 (u1 )λ21 (l(u))−1 1{u6=0} P (du). It is easy to see that limE 3ε↓0 limk→∞ Ik, ε = 0. Step 3: Fix ε ∈ E such that 0 < ε < 1. We introduce the non-negative numbers Θk := %k + |βk,1 | for k ≥ 1 and consider two cases as follows. If lim infk→∞ Θk = 0, in this case R(λ1 , λ2 ) = 0. If lim infk→∞ Θk > 0, there exists a subsequence, denoted again by {Θk }, converging to Θ ∈ (0, ∞]. Then the following limits exist (passing to a subsequence if necessary)

R




%k βk,1 (i) → % ∈ [0, 1], → β1 ∈ [−1, 1], ak, ε → a(εi) ∈ [−1, 1], Θk Θk Z Z 1 χ (u1 ) Pk (du) → c0 ∈ [0, 1/ε 2 ], Pk (du) → c1 ∈ [−1/ε 2 , 1/ε 2 ], G\Q l(u) G\Q l(u)

(3.5)

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C. Ma / Statistics and Probability Letters 79 (2009) 1710–1716

for i = 1, 2. If % = 0, then we have get in the limit:

Z

R(λ1 , λ2 ) = β1 λ1 . Now suppose that % > 0. Dividing both sides of Eq. (3.4) by Θk we

e−hλ,ui (l(u))−1 Pk (du) = L(λ),

lim

k→∞

1

Θ

(3.6)

G \Q

where L(λ) is some function of λ and is continuous at λ = 0. Since (l(u))−1 tends to 1 as u → ∆ and is the bounded continuous function of u ∈ G∆ \ Q , we have

Z G ∆ \Q

Z

(l(u))−1 P (du) = lim

Z

k→∞

(l(u))−1 P (du) = lim

Z

n→∞

G \Q

(l(u))−1 Pk (du) = L(0).

(3.7)

G\Q 1

G∆ \Q

e−h n , ui (l(u))−1 P (du) = lim L

 

n→∞

1

n

= L(0).

(3.8)

It follows from (3.7) and (3.8) that P ({∆}) = 0. Thus the subsequence {Pk } converges to the probability P on G and P is supported by R2+ . Define v({0}) = 0 and v(du) = (l(u))−1 P (du) on {u ∈ R2+ : kuk > 0}; it is not hard to show that

Z lim lim Jk, ε =

E 3ε↓0 k→∞

R2+

h(u, λ)v(du),

(i) (i) lim lim inf ak, ε = lim lim sup ak, ε = a(i) ,

E 3ε↓0 k→∞

E 3ε↓0

(3.9)

k→∞

for i = 1, 2 and a(i) ≥ 0. Then we obtain that 1

Θ

R(λ1 , λ2 ) = β1 λ1 + %a

(2)

(1) 2

λ2 − %a λ1 − %

Z R2+

h(u, λ)v(du).

(3.10)

Step 4: Now we need to verify that 1/Θ > 0. If not, the r.h.s. of (3.10) is the zero function in λ. Assume β1 = 0
and % v R2+ \ {0} = 0. Then % = 1. Hence we have P (R2+ \ {0}) = 0 and then P ({0}) = 1. (3.9) shows that 2a(1) + a(2) = limE 3ε↓0 P (kuk ≤ ε) = 1. Thus a(i) cannot be zero at the same time for i = 1, 2. But the representation of the function in λ on the r.h.s. of (3.10) with parameters (β1 , (% a(i) ), % ν) is unique; see, e.g., Theorem 8.1 of Sato (1999). Then we must have that a(i) = 0 for i = 1, 2. Therefore 1/Θ = 0 is impossible. Let aˆ 11 = Θ β1 , a12 = %Θ a(2) , α = % Θ a(1) and µ1 (·) = % Θ v(·). Then R(λ1 , λ2 ) = aˆ 11 λ1 + a12 λ2 − αλ − 2 1

Z

e−hλ,ui − 1 + λ1 χ (u1 ) µ1 (du).




R2

+

((

+ u2 ) ∧ 1) µ1 (du) < ∞. By applying dominated convergence to the above formula, we have for λ1 > 0, Z Z R01 (λ1 , 0) = aˆ 11 − µ1 ({u1 > 1}) − 2αλ1 − u1 (1 − e−λ1 u1 )µ1 (du) + u1 e−λ1 u1 µ1 (du);

Note that

R

R2+

u21

{0
{u1 >1}

for λ2 > 0, R02 (0, λ2 ) = a12 +

Z R2

u2 e−λ2 u2 µ1 (du).

+

Applying dominated convergence and monotone convergence to the above two formulas, we get R01 (0+, 0) = aˆ 11 − µ1 ({u1 > 1}) + R02 (0, 0+) = a12 +

Z R2+

Z {u1 >1}

u1 µ1 (du),

u2 µ1 (du).

(u1 − R2+ χ(u1 ))µ1 (du). Then we obtain (1.1). a (ii) Firstly, suppose |a11 | + a12 > 0. If a11 ≤ 0 and a12 ≥ |a11 |, set γ1,k = a12 and g1,k (λ1 , λ2 ) = λ1 λ2 − a11 (1 − λ1 )λ2 ; if 12 a11 a11 ≥ 0 and a12 ≥ |a11 |, set γ1,k = a12 and g1,k (λ1 , λ2 ) = λ1 λ2 + a λ1 λ2 (λ1 − 1); if a11 ≤ 0 and |a11 | ≥ a12 , set γ1,k = |a11 | 12 a a and g1,k (λ1 , λ2 ) = 1 + a12 (1 − λ2 ); if a11 ≥ 0 and |a11 | ≥ a12 , set γ1,k = |a11 | and g1,k (λ1 , λ2 ) = λ21 + a12 λ21 (λ2 − 1). 11 11 α 2 Secondly, set γ2,k = (2α + 1)k and g2,k (λ1 , λ2 ) = λ1 + 2α+1 (1 − λ1 ) . Thirdly, suppose that µ1 6= 0. Let Dk = {(u1 , u2 ) ∈ From the fact that R is locally Lipschitz on R2+ , we have

R

R2+

l12 (u1 ) + l1 (u2 ) µ1 (du) < ∞. Let a11 = aˆ 11 +




R

C. Ma / Statistics and Probability Letters 79 (2009) 1710–1716

1715

√ √ R
R2+ : u1 > 1/ k, u2 > 1/ k} and σk = D u1 − 1/k µ1 (du). Set the sequences k γ3,k = σk + µ1 (Dk )/k + 1,   Z σk + 1 σk 1 −kh1−λ,ui e µ1 (du) + λ1 + (1 − λ1 ) . g3,k (λ1 , λ2 ) = kγ3,k Dk γ3,k σk + 1 (1)

= γ1,k + γ2,k + γ3,k and let gˆk(1) = γk−1 (γ1,k g1,k + γ2,k g2,k + γ3,k g3,k ). Then the sequence {R(k1) } with (γˆk , gˆk ) satisfies (A) and (B). In a similar way, we also have the sequence {R(k2) } with (γˆk(2) , gˆk(2) ) satisfying (A) and (B). (1) (2) (1) (1) (1) (2) (2) (2) (2) (1) Now let γk = γˆk + γˆk , gk (λ1 , λ2 ) = γk−1 (γˆk gˆk (λ1 , λ2 ) + γˆk λ1 ), and gk (λ1 , λ2 ) = γk−1 (γˆk gˆk (λ1 , λ2 ) + γˆk λ2 ). (l) (l) Then we find the common sequences {γk } and {gk } such that (A) and (B) hold with Rk → R(l) for l = 1, 2.  Finally, we let γˆk (1)

(1)

Proof of Proposition 2.2. (i) Let [a, b]2 be any bounded rectangle in R2+ . Assume {(xk , yk ), k ≥ 0} ⊆ [a, b]2 , and xk → x0 and yk → y0 . There exists η > 0 such that

|F (x0 + η, y0 + η) − F (x0 , y0 )| ∨ |F (x0 − η, y0 − η) − F (x0 , y0 )| < ε, because F is continuous. There is k0 such that if k ≥ k0 , we have |xk − x0 | ∨ |yk − y0 | < η, and

|Fk (x0 + η, y0 + η) − F (x0 + η, y0 + η)| ∨ |Fk (x0 − η, y0 − η) − F (x0 − η, y0 − η)| < ε, since Fk → F pointwise. Note that for fixed k, Fk (λ1 , λ2 ) is nondecreasing in separate λi (i = 1, 2). Then we have for k ≥ k0 , F (x0 , y0 ) − 2ε ≤ Fk (x0 − η, y0 − η) ≤ Fk (x0 , y0 ) ≤ Fk (x0 + η, y0 + η) ≤ F (x0 , y0 ) + 2ε.

˜ k (λ1 , λ2 ) = γk 1 − hk (e−λ1 /k , e−λ2 /k ) . By the above argument, we have that Thus Fk → F locally uniformly. Set Φ 



˜ k (λ1 , λ2 ) → F (λ1 , λ2 ). Hence it suffices to consider the limit representation of Φ ˜ k . Continuing as in the proof of Φ Proposition 2.1, we have (1.3). (ii) Firstly suppose that b1 + b2 > 0. Set γ1,k = k(b1 + b2 ) and h1,k (λ1 , λ2 ) =

b1 λ + b1b+2b2 λ2 . Secondly, b1 +b2 1 √ √ suppose that m 6= 0. Let Dk = {(u1 , u2 ) ∈ R2+ : u1 > 1/ k, u2 > 1/ k }. Define the sequences γ2,k = m(Dk ) and R h2,k (λ1 , λ2 ) = m(Dk )−1 D e−kh1−λ,ui m(du). Finally, we let γk = γ1,k + γ2,k and let hk = γk−1 (γ1,k h1,k + γ2,k h2,k ). Then the k sequence {Fk } defined by (2.2) satisfies (A) and (C). 

4. Proof of Theorem 2.1 Proof of Theorem 2.1. (i) Let (Pt )t ≥0 denote the transition semigroup of the two-dimensional CBI-process corresponding to ((R(1) , R(2) ), F ) and let A be the infinitesimal generator. For λ = (λ1 , λ2 )  0 and x = (x1 , x2 ) ∈ R2+ , set eλ (x) = e−hλ,xi . Then we have Aeλ (x) = −ehλ,xi x1 R(1) (λ) + x2 R(2) (λ) + F (λ) .





(4.1)

Denote by D1 the linear hull of {eλ , λ  0}. Then D1 is an algebra which strongly separates the points of R2+ . Let C0 (R2+ ) be the space of continuous functions on R2+ vanishing at infinity. By the Stone–Weierstrass theorem, D1 is dense in C0 (R2+ ) for the supremum norm. Note that D1 is invariant under  (Pt )t ≥0 by (1.4). It follows from
Proposition 3.3 in Chapter 1 of Ethier and Kurtz (1986) that D1 is the core of A. Note that Yk (n)/k = y1,k (n)/k, y2,k (n)/k , n ≥ 0 is a Markov chain with state space Ek := {(i/k, j/k) : (i, j) ∈ N2 } and one-step transition probability Qk (x, dy) determined by

Z

(1)


kx1

e−hλ,yi Qk (x, dy) = gk (e−λ1 /k , e−λ2 /k )

(2)


kx2

gk (e−λ1 /k , e−λ2 /k )

h(e−λ1 /k , e−λ2 /k ).

Ek

Then the (discrete) generator Ak of {Yk ([γk t ])/k, t ≥ 0} is given by Ak eλ (x) = γk

h

(1)


kx1

gk (e−λ1 /k , e−λ2 /k )

" =e

−hλ,xi

(

γk exp −

2 X

(l)

(2)


kx2

gk (e−λ1 /k , e−λ2 /k )

xl αk (λ)e

λl /k (l)

)

hk (e−λ1 /k , e−λ2 /k ) − e−hλ,xi

n

o

#

Sk (λ) exp −βk (λ)S˜k (λ) − 1 ,

l =1

where


−1
(l) αk(l) (λ) = eλl /k gk(l) (e−λ1 /k , e−λ2 /k ) − 1 log eλl /k gk (e−λ1 /k , e−λ2 /k ) ,   (l) Sk (λ1 , λ2 ) = k e−λl /k − gk (e−λ1 /k , e−λ2 /k ) ,
−1
βk (λ) = hk (e−λ1 /k , e−λ2 /k ) − 1 log hk (e−λ1 /k , e−λ2 /k ) , S˜k (λ1 , λ2 ) = 1 − hk (e−λ1 /k , e−λ2 /k ),

i

1716

C. Ma / Statistics and Probability Letters 79 (2009) 1710–1716

(l)

(l)

˜ k (λ1 , λ2 ) = γk S˜k (λ1 , λ2 ). By conditions (A), (B), (C) and the proof of for l = 1, 2. Let Φk (λ1 , λ2 ) = γk Sk (λ1 , λ2 ) and Φ Proposition 2.1 (see (3.1)), it is easy to show that for l = 1, 2, (l) lim Φk (λ1 , λ2 ) = R(l) (λ1 , λ2 ),

˜ k (λ1 , λ2 ) = F (λ1 , λ2 ), lim Φ

k→∞

(l)

k→∞

(l)

lim eλl /k gk (e−λ1 /k , e−λ2 /k ) − 1 = 0,

lim αk (λ) = 1,




k→∞

lim hk (e−λ1 /k , e−λ2 /k ) − 1 = 0,

k→∞

lim βk (λ) = 1.




k→∞

(4.2) (4.3) (4.4)

k→∞

Then we have Ak eλ (x) = −e

−hλ,xi

" 2 X

(l)

xl ak (λ)e

λl /k

#

(l)

˜ k (λ) + o(1). Φk (λ) + βk (λ)Φ

(4.5)

l =1

In view of (4.2)–(4.5), we obtain limk→∞ supx∈Ek |Ak eλ (x) − Aeλ (x)| = 0. From Corollary 8.9 in Chapter 4 of Ethier and Kurtz (1986), we prove the first part of Theorem 2.1. (1) (2) (ii) By Proposition 2.1, we have the sequences {γ¯k }, {¯gk } and {¯gk } such that (A) and (B) hold. By Proposition 2.2, we (1) (1) also have {γ˜k } and {h˜ k } such that (A) and (C) hold. Now let γk = γ¯k + γ˜k , gk (λ1 , λ2 ) = γk−1 (γ¯k g¯k (λ1 , λ2 ) + γ˜k λ1 ), (2)

(2)

gk (λ1 , λ2 ) = γk−1 (γ¯k g¯k (λ1 , λ2 ) + γ˜k λ2 ), and hk (λ1 , λ2 ) = γk−1 (γ˜k h˜ k (λ1 , λ2 ) + γ¯k ). Then we get the common sequences

{γk }, {gk }, {gk }, and {hk } such that (A), (B) and (C) hold with R(k1) → R(1) , R(k2) → R(2) , and Fk → F . Thus the second part (1)

(2)

of Theorem 2.1 follows from Theorem 2.1(i).



Acknowledgements We thank the referee for pointing out some mistakes in an earlier version of the paper. The author is supported by NSFC (No. 10871103). References Duffie, D., Filipović, D., Schachermayer, W., 2003. Affine processes and applications in finance. Ann. Appl. Probab. 13, 984–1053. Ethier, S.N., Kurtz, T.G., 1986. Markov Processes: Characterization and Convergence. John Wiley and Sons Inc., New York. Kawazu, K., Watanabe, S., 1971. Branching processes with immigration and related limit theorems. Theor. Probab. Appl. 16, 36–54. Li, Z.H., 1991. Integral representations of continuous functions. Chinese Sci. Bull. (English Ed.) 36, 976–983. Preprint form available at:math.bnu.edu.cn/~lizh. Li, Z.H., 2006a. A limit theorem of discrete Galton–Watson branching processes with immigration. J. Appl. Probab. 43, 289–295. Li, Z.H., 2006b. Branching processes with immigration and related topics. Front. Math. China 1, 73–97. Li, Z.H., Ma, C.H., 2008. Catalytic discrete state branching models and related limit theorems. J. Theor. Probab. 21, 936–965. Pitman, J., Yor, M., 1982. A decomposition of Bessel bridges. Z. Wahrscheinlichkeitstheor verwandte Geb. 59, 425–457. Sato, K., 1999. Lévy processes and infinitely divisible distributions. In: Cambridge Studies in Advanced Mathematics, vol. 68. Cambridge University Press, Cambridge. Shiga, T., Watanabe, S., 1973. Bessel diffusions as a one-parameter family of diffusion processes. Z. Wahrscheinlichkeitstheor verwandte Geb. 27, 37–46. Venttsel’, A.D., 1959. On boundary conditions for multi-dimensional diffusion processes. Theor. Probab. Appl. 4, 164–177. Watanabe, S., 1969. On two dimensional Markov processes with branching property. Trans. Amer. Math. Soc. 136, 447–466.