Critical branching processes with immigration

10Critical branching processes with immigration

Marton Ispany and Gyula Pap

Abstract In this paper we give a survey on some results concerning critical andnearly critical Galton–Watson branching processes with immigration. As a byprod-uct of a general limit theorem for weak convergence of step processes of mar-tingale differences towards a diffusion process, functional limit theorems can beproved for different models. The limit process is either a squared Bessel process oran Ornstein–Uhlenbeck type process. The asymptotic behavior of conditional leastsquares estimator of the offspring mean will also be described. The results are ap-plied in the theory of integer-valued autoregression as well.

Mathematics Subject Classification (2000): 60J80, 60F17, 62F12

Keywords: critical branching process with immigration, conditional least squaresestimator.

10.1 Introduction

The theory of branching processes allowing immigrants joining to the populationin each generation has been studied for a long time, see, e.g., Sevastyanov [26]and Harris [8]. The limit distribution of a branching process with immigration hasbeen described by Heathcote [9] and Foster [4] in the most elementary cases. Theyproved that if the offspring mean is greater than or equal to 1, i.e., the model is su-percritical or critical, then the process tends to infinity, while if the offspring meanis less than 1, i.e., the model is subcritical, and the immigration mean is finite then

Marton IspanyFaculty of Informatics, University of Debrecen, Pf.12, Debrecen H-4010, Hungary,e-mail: [email protected]

Gyula PapFaculty of Informatics, University of Debrecen, Pf.12, Debrecen H-4010, Hungary,e-mail: [email protected]

M.G. Velasco et al. (eds.), Workshop on Branching Processes and Their Applications, 135Lecture Notes in Statistics – Proceedings 197, DOI 10.1007/978-3-642-11156-3 10,c© Springer-Verlag Berlin Heidelberg 2010

136 Marton Ispany and Gyula Pap

the process converges weakly to the (unique) stationary distribution. A necessaryand sufficient condition for convergence in distribution to a proper random variablehas been proved by Foster and Williamson [5], see Athreya and Ney [2, TheoremVI.7.2]. Moreover, in the supercritical case Seneta [24] proved that under appropri-ate normalization the process converges almost surely to a random variable. Finally,in the critical case Foster [4] and Seneta [25] showed that the process normalizedby the number of generation converges in distribution to a gamma distribution.

In this paper critical and nearly critical Galton–Watson branching processes withimmigration are investigated and related functional limit theorems are presented.That is, we prove not only the weak convergence of the one dimensional distribu-tions but the weak convergence of finite dimensional distributions and tightness.Our technique is the martingale method and the proofs are based on a generalconvergence theorem for martingale differences, see Theorem 10.1. The first suchtheorem in the critical case has been proved by Wei and Winnicki [29, 30], seeTheorem 10.2.

The paper is organized as follows. In Sect. 10.2 a two-way connection is pre-sented between the branching processes with immigration and conditionally het-eroscedastic autoregressive processes. Section 10.3 is devoted to the main functionallimit theorem with application to the Wei–Winnicki’s theorem. In Sect. 10.4 nearlycritical branching processes with immigration are considered. Finally, in the lastsection, as an application, the asymptotic behaviour of the conditional least squaresestimator of the offspring mean is investigated under various assumptions.

10.2 Branching and autoregressive processes

Let {ξk, j, εk : k, j ∈ N} be independent, non-negative, integer-valued random vari-ables such that {ξk, j : k, j ∈ N} and {εk : k ∈ N} are identically distributed. Definerecursively

Xk =Xk−1

∑j=1

ξk, j + εk for k ∈ N, X0 = 0. (10.1)

The sequence (Xk)k∈Z+ is called a branching process with immigration. We caninterpret Xk as the size of the kth generation of a population, where ξk, j is thenumber of offsprings of the jth individual in the (k − 1)st generation and εk isthe number of immigrants contributing to the kth generation. Assume that

m := Eξ1,1 < ∞, λ := Eε1 < ∞, σ2 := Varξ1,1 < ∞, b2 := Varε1 < ∞.

The cases m < 1, m = 1 and m > 1 are referred to respectively as subcritical,critical and supercritical. Such processes have a number of applications in biology,finance, economics, queueing theory etc., see e.g. Haccou et al. [7].

In particular, if the offspring distribution is a Bernoulli distribution then thebranching process with immigration is called first order integer-valued autoregres-

10 Critical branching processes with immigration 137

sive (INAR(1)) time series. It has been introduced by Al–Osh and Alzaid [1]. AnINAR(1) process may also be written in the form

Xk = m◦Xk−1 + εk for k ∈ N, X0 = 0, (10.2)

where we use the thinning or Steutel and van Harn operator m◦ which is definedfor m ∈ [0,1] and for a non-negative integer-valued random variable X by

m◦X :=

⎧⎨

⎩

X∑j=1

ξ j, X > 0,

0, X = 0,

where the counting sequence (ξ j) j∈N consists of independent and identically dis-tributed Bernoulli random variables with mean m, independent of X (see Steuteland van Harn [28]), and the counting sequences involved in m ◦Xk−1, k ∈ N, aremutually independent and independent of (εn)n∈N. Formula (10.2) shows the anal-ogy with the common AR(1) process, i.e., m plays the role of an autoregressiveparameter and {εk : k ∈ N} is an innovation or driving process. Motivation in or-der to include discrete data models comes from the need to account for the discretenature of certain data sets, often counts of events, objects or individuals. Branchingprocess with immigration is a promising model to describe such phenomenas. Ex-amples of applications can be found in review papers by McKenzie [20], Jung andTremayne [19], and Weiß [32].

For k ∈ Z+, let Fk denote the σ -algebra generated by X0,X1, . . . ,Xk. Then,by (10.1), we have the conditional expectation

E(Xk | Fk−1) = mXk−1 +λ , k ∈ N.

Clearly,Mk := Xk −E(Xk | Fk−1) = Xk −mXk−1 −λ , k ∈ N, (10.3)

defines a martingale difference sequence (Mk)k∈N with respect to the filtration(Fk)k∈Z+ , thus we have the recursion

Xk = λ +mXk−1 +Mk for k ∈ N, X0 = 0. (10.4)

Hence a branching process with immigration can be rewritten as an autoregres-sive process with drift, where the driving process (Mk)k∈N is a sequence of mar-tingale differences. The main difference from the common autoregressive processis in the nature of conditional variance. An AR(1) process is homoscedastic, i.e.,E(M2

k | Fk−1) is constant for all k ∈ N. Contrarily, the autoregressive representa-tion of branching processes with immigration has heteroscedastic conditional struc-ture. Namely, for the conditional variance we have E(M2

k | Fk−1) = σ2Xk−1 + b2

since, by (10.3) and (10.1),


Mk = Xk −mXk−1 −λ =Xk−1

∑j=1

(ξk, j −m)+(εk −λ ).

Summarizing, there is a natural two-way connection between the branching pro-cesses with immigration and autoregressive processes. On one hand, branching pro-cess with immigration is a useful alternative to model integer-valued time series.On the other hand, a branching process with immigration is an autoregressive pro-cess with conditionally heteroscedastic innovation. Thus, to prove functional limittheorem for branching processes with immigration we may apply limit theoremsdeveloped for heteroscedastic autoregressive processes and more general stochasticprocesses.

10.3 Functional limit theorems

The functional limit theorems of this section are based on a general limit theorem formartingale differences. For each n ∈ N, let (Un

k )k∈N be a sequence of Rd-valued

square-integrable martingale differences with respect to a filtration (F nk )k∈Z+ , i.e.,

(i) Unk is F n

k -measurable for all k,n ∈N, and (ii) E‖Unk ‖2 <∞, E

(Un

k |F nk−1

)= 0

for all k,n ∈ N. Introduce the random step functions

U nt :=

�nt�

∑k=1

Unk , t ∈ R+, n ∈ N.

Moreover, let (Ut)t∈R+ be a (not necessarily time-homogeneous) d-dimensionaldiffusion process with zero drift, i.e.,

dUt = γ(t,Ut)dWt , t ∈ R+,

where γ : R+ × Rd → R

d×r is a continuous function and (Wt)t∈R+ is an r-dimensional standard Wiener process. Assume that the SDE has a unique weaksolution with U0 = x0 for all x0 ∈ R

d . Let (Ut)t∈R+ be a solution with U0 = 0.Our martingale limit theorem is derived from a general semimartingale limit the-

orem due to Jacod and Shiryayev [18, Theorem IX.3.39], but the assumptions of thefollowing theorem are much easier to verify.

Theorem 10.1. Suppose that for each T > 0,

(i) supt∈[0,T ]

∥∥∥∥∥

�nt�∑

k=1E(Un

k (Unk )� | F n

k−1

)−∫ t

0 γ(s,U ns )γ(s,U n

s )�ds

∥∥∥∥∥

P−→ 0,

(ii)�nT�∑

k=1E(‖Un

k ‖21{‖Unk ‖>θ} | F n

k−1

) P−→ 0 for all θ > 0.

ThenU n D−→ U as n → ∞,


that is, weakly in the Skorokhod space D(R+,Rd).

We note that in assumption (i) uniform convergence on compacts in probability isinvolved and assumption (ii) is the conditional Lindeberg condition. The proof ofthis theorem can be found in Ispany and Pap [12, 13]. In the sequel, we check onlyassumption (i) in the proof of limit theorems. Assumption (ii) can be verified in thesame manner, see the cited reference in each case.

The celebrated Wei–Winnicki’s theorem, see [29] and [30], describes the asymp-totic behaviour of a critical branching process with immigration. Introduce the ran-dom step functions

X nt := X�nt� for t ∈ R+, n ∈ N.

Theorem 10.2. For a critical branching process with immigration we have

n−1X n D−→ X as n → ∞, (10.5)

where (Xt)t∈R+ is the solution of the stochastic differential equation

dXt = λ dt +σ√

(Xt)+ dWt , t ∈ R+, X0 = 0, (10.6)

where x+ := max{x,0}, and (Wt)t∈R+ is a standard Wiener process.

It is well known that the SDE (10.6) has a unique global strong solution such thatXt ≥ 0 almost surely for all t ∈ R+. Thus, one may replace (Xt)+ by Xt underthe square root. (See, e.g., Ikeda and Watanabe [10, Example IV.8.2].) The process(Xt)t∈R+ is a continuous branching process called square-root process, squaredBessel process, or Cox–Ingersoll–Ross model.

Proof. We apply the martingale limit theorem, Theorem 10.1, with the choice γ(x, t):=σ

√(x+λ t)+, and prove that M n D−→ M as n →∞, where M n

t := 1n ∑

�nt�k=1 Mk

and dMt = σ√

(Mt +λ t)+ dWt , t ∈ R+, M0 = 0. Indeed, by (10.4), Xk =Xk−1 +λ +Mk implies Xk =∑k

j=1(Mj +λ ). Hence the following heuristics provescondition (i). The conditional covariances admit the asymptotics

1n2

�nt�

∑k=1

E(M2k | Fk−1) =

1n2

�nt�

∑k=1

(σ2Xk−1 +b2) ≈ σ2

n2

�nt�

∑k=1

k−1

∑j=1

(Mj +λ )

=σ2

n

�nt�

∑k=1

(M n

k/n +λk−1

n

)≈ σ2

∫ t

0(M n

s +λ s)ds.

The conditional Lindeberg condition (ii) can be verified as in Ispany [11] or Ispanyet al. [16]. Finally, the continuous mapping theorem proves (10.5) since

1n

X�nt� =1n

�nt�

∑j=1

(Mj +λ ) D−→ Mt +λ t = Xt as n → ∞. ��


10.4 Nearly critical branching processes with immigration

In this section we study branching processes with immigration which are close tothe criticality. To be precise we consider a sequence of branching processes withimmigration (Xn

k )k∈Z+ , n ∈ N, given by the recursion

Xnk =

Xnk−1

∑j=1

ξ nk, j + εn

k for k, n ∈ N, Xn0 = 0, (10.7)

where {ξ nk, j, ε

nk : k, j, n ∈ N} are independent, nonnegative, integer-valued random

variables such that {ξ nk, j : k, j ∈ N} and {εn

k : k ∈ N} for each n ∈ N are identicallydistributed. Assume furthermore that, for all n ∈ N,

mn := Eξ n1,1 < ∞, λn := Eεn

1 < ∞, σ2n := Varξ n

1,1 < ∞, b2n := Varεn

1 < ∞.

Definition 10.1. A sequence of branching processes with immigration is callednearly critical with rate α ∈ R if mn = 1+αn−1 +o(n−1) as n → ∞.

This kind of the parametrization of the offspring mean has been considered bySriram [27] for the first time. The notion of nearly criticality or nearly unstability hasbeen suggested by Chan and Wei [3] in case of AR(1) models. The main motivationcomes from the econometrics, where the so-called “unit-root problem” plays animportant role.

The following theorem, see Ispany [11, Theorem 2.1] is a generalization of theWei–Winnicki’s theorem and Sriram’s theorem, see [27, Theorem 3.1]. In the limittheorem we apply a kind of “self-normalization”, namely we divide by the off-spring variance. Such kind of normalization is investigated recently by Rahimov[21], where the offspring variance is modelled by a slowly varying function. Intro-duce the random step functions

X nt := Xn

�nt�, M nt :=

�nt�

∑k=1

Mnk for t ∈ R+, n ∈ N.

Theorem 10.3. Suppose that σ2n > 0 for all n ∈ N, and

(i) E(|ξ n

1,1 −mn|21{|ξ n1,1−mn|>θnσ2

n }

)= o(σ2

n ) as n → ∞ for all θ > 0,

(ii) λn = λσ2n +o(σ2

n ) as n → ∞ for some λ ≥ 0,(iii) b2

n = o(nσ4n ) as n → ∞.

Then(nσ2

n

)−1EX n

t → λ∫ t

0eαs ds as n → ∞ (10.8)

for all t ∈ R+, and

(nσ2

n

)−1X n D−→ X as n → ∞,


that is, weakly in the Skorokhod space D(R+,R), where(Xt)

t∈R+is the unique

solution of the stochastic differential equation (SDE)

dXt = (λ +αXt)dt +√

(Xt)+ dWt , t ∈ R+, X0 = 0, (10.9)

where (Wt)t∈R+ is a standard Wiener process.

If the offspring variance tends to 0, e.g., in case of Bernoulli offspring distri-bution, then the above theorem gives a trivial deterministic limit process. How-ever, in this case going one step further a fluctuation theorem holds with Ornstein–Uhlenbeck type limit process, see Ispany et al. [16, Theorem 2.2].

Theorem 10.4. Suppose that

(i) σ2n = βn−1 +o(n−1) as n → ∞ with some β ≥ 0,

(ii) nE(|ξ n

1,1 −mn|21{|ξ n1,1−mn|>θ

√n}

)→ 0 as n → ∞ for all θ > 0,

(iii) λn → λ and b2n → b2 as n → ∞ with some λ ≥ 0 and b2 ≥ 0,

(iv) E(|εn

1 −λn|21{|εn1 −λn|>θ

√n}

)→ 0 as n → ∞ for all θ > 0.

Thenn−1/2 (X n −EX n, M n) D−→

(X ,M

)as n → ∞,

that is, weakly in the Skorokhod space D(R+,R2), where(Mt

)

t∈R+is a time-

changed Wiener process, namely, Mt = W (Tt), t ∈ R+ with

Tt :=∫ t

0ρ(s)ds, ρ(t) := b2 +βλ

∫ t

0eαs ds, t ∈ R+,

(W (t))t∈R+ is a standard Wiener process, and

Xt :=∫ t

0eα(t−s) dMs, t ∈ R+,

is an Ornstein–Uhlenbeck type process driven by (Mt)t∈R+ .

A more general approximation theorem has been proved in Ispany et al. [17] forOrnstein–Uhlenbeck processes using sequence of branching processes with immi-gration.

Finally, we may investigate the nearly critical behaviour in the framework ofone model only allowing inhomogeneous parameters, i.e., considering branchingprocesses with immigration in varying environment. The next theorem, see Gyorfi etal. [6, Theorem 2], shows that if the convergence to criticality is slow, then the limitis a Poisson distribution and we do not need any normalization. The inhomogeneousINAR(1) process (Xn)n∈Z+ is defined by

Xk = mk ◦Xk−1 + εk for k ∈ N, X0 = 0,


where {εk : k ∈ N} are non-negative integer-valued random variables with λk :=Eεk < ∞ and b2

k := Varεk < ∞.

Theorem 10.5. Assume that

(i) mn < 1 for all n ∈ N, limn→∞

mn = 1,∞∑

n=1(1−mn) = ∞,

(ii) limn→∞

λn1−mn

= λ ∈ [0,∞), limn→∞

b2n

1−mn= 0.

ThenXn

D−→ Po(λ ) as n → ∞.

The proof of this theorem is based on Poisson approximation.

10.5 Conditional least squares estimators

Let us consider the branching process (10.1). The conditional least squares estimator(CLSE) mn of m based on the observations X1, . . . ,Xn assuming that λ is knowncan be obtained minimizing the sum of squares

n

∑k=1

(Xk −mXk−1 −λ )2

with respect to m, and it has the form

mn = ∑nk=1(Xk −λ )Xk−1

∑nk=1 X2

k−1

,

hence

mn −m = ∑nk=1 MkXk−1

∑nk=1 X2

k−1

.

Theorem 10.6. For a subcritical branching process with immigration under the as-sumptions Eξ 3

1,1 < ∞, Eε31 < ∞ we have

n1/2(mn −m) D−→ N (0,σ2sub)

with

σ2sub :=

σ2∑∞j=0 j3 p j +b2∑∞

j=0 j2 p j(∑∞

j=0 j2 p j

)2 ,

where (p j) j∈Z+ denotes the unique stationary distribution of the Markov chain(Xk)k∈Z+ .


Proof. First observe ∑nk=1 X2

k−1 → ∑∞j=0 j2 p j a.s. by the Ergodic Theorem. Then

by the Martingale Central Limit Theorem we obtain n−1/2∑�nt�k=1 MkXk−1

D−→ cWt ,where (Wt)t∈R+ is a standard Wiener process and c2 := σ2 ∑∞

j=0 j3 p j + b2∑∞j=0 j2 p j .

Indeed, the conditional covariances admit the asymptotics

1n

�nt�

∑k=1

E(M2k X2

k−1 | Fk−1) =1n

�nt�

∑k=1

(σ2Xk−1 +b2)X2k−1 ≈ c2t

again by the Ergodic Theorem. ��

The asymptotic behaviour of the CLSE of the offspring mean is unknown if the off-spring or immigration distribution have infinite third moment. Simulations suggestthat the limit distribution is not normal, we suspect it is a stable distribution.

Wei and Winnicki [29, 31] described the asymptotic behavior of the CLSE in thecritical case m = 1 with σ2 > 0.

Theorem 10.7. For a critical branching process with immigration under the as-sumption σ2 > 0 we have

n(mn −1) D−→∫ 1

0 Xt d(Xt −λ t)∫ 1

0 X 2t dt

,

where the process (Xt)t∈R+ is given in Theorem 10.2.

Proof. This can be proved using the general martingale limit theorem, Theorem10.1. For each n ∈ N, consider the martingale differences

Unk :=

[n−1Mk

n−2MkXk−1

]= Mk

[n−1

n−2Xk−1

], k ∈ N,

with respect to the filtration F nk := σ(M1, . . . ,Mk). The conditional covariances

admit the asymptotics

�nt�

∑k=1

E[Unk (Un

k )� | F nk−1] =

�nt�

∑k=1

E(M2k | Fk−1)

[n−1

n−2Xk−1

][n−1

n−2Xk−1

]�

=�nt�

∑k=1

(σ2Xk−1 +b2)[

n−2 n−3Xk−1

n−3Xk−1 n−4X2k−1

]≈∫ t

0γ(s,U n

s )γ(s,U ns )�ds

with γ : R+ × R2 → R

2×1, γ(

s,

[xy

])= σ

[(x+λ s)1/2

+

(x+λ s)3/2+

]

, since (10.4) yields

Xk =k∑j=1

(Mj +λ ) =k∑j=1

Mj + kλ .

By Theorem 1 we obtain(

n−1∑�nt�k=1 Mk, n−2∑�nt�

k=1 MkXk−1

)D−→ (Mt ,Yt), where


[dMt

dYt

]= γ(

t,

[Mt

Yt

])dWt = σ

[(Mt +λ t)1/2

+

(Mt +λ t)3/2+

]

dWt , t ∈ R+,

M0 = Y0 = 0. Hence

dMt = σ(Mt +λ t)1/2+ dWt ,

Yt = σ∫ t

0(Ms +λ s)3/2

+ dWs =∫ t

0(Ms +λ s)+ dMs.

We also obtain n−1X�nt� = n−1∑�nt�j=1(Mj +λ ) D−→ Mt +λ t = Xt (compare with

Theorem 10.2). Consequently Yt =∫ t

0 Xs dMs and

(

n−3�nt�

∑k=1

X2k−1, n−2

�nt�

∑k=1

MkXk−1

)D−→(∫ t

0X 2

s dt,∫ t

0Xs dMs

),

hence

n(mn −1

)=

n−2∑nk=1 Xk−1Mk

n−3∑nk=1 X2

k−1

D−→∫ 1

0 Xt dMt∫ 1

0 X 2t dt

as n → ∞. ��

The critical case m = 1 with σ2 = 0 has been described by Ispany et al. [14].

Theorem 10.8. For a critical branching process with immigration under the as-sumptions σ2 = 0 and λ > 0 we have

n3/2 (mn −1) D−→ N (0,σ2crit) with σ2

crit :=3b2

λ 2 .

Proof. This can be proved again using Theorem 10.1. For each n ∈ N, considernow the martingale differences

Unk :=

[n−1/2Mk

n−3/2MkXk−1

]= Mk

[n−1/2

n−3/2Xk−1

], k ∈ N,

with respect to the filtration F nk := σ(M1, . . . ,Mk). The conditional covariances

admit the asymptotics

�nt�

∑k=1

E[Unk (Un

k )� | F nk−1] =

�nt�

∑k=1

E(M2k | Fk−1)

[n−1/2

n−3/2Xk−1

][n−1/2

n−3/2Xk−1

]�

=�nt�

∑k=1

b2[

n−1 n−2Xk−1

n−2Xk−1 n−3X2k−1

]≈∫ t

0γ(s,U n

s )γ(s,U ns )�ds


with γ : R+ ×R2 → R

2×1, γ(

s,

[xy

])= b

[1

λ s

]

, since now Xk = Xk−1 + εk =

k∑j=1

ε j yields n−1X�nt� = n−1�nt�∑j=1

ε jD−→ λ t =: Xt . By Theorem 10.1 we obtain

(n−1/2∑�nt�

k=1 Mk, n−3/2∑�nt�k=1 MkXk−1

)D−→ (Mt ,Yt), where

[dMt

dYt

]= γ(

t,

[Mt

Yt

])dWt = b

[1

λ t

]

dWt , t ∈ R+,

M0 = Y0 = 0. Thus Mt = bWt and Yt = bλ∫ t

0 sdWs =∫ t

0 Xs dMs, and(n−3∑�nt�

k=1 X2k−1, n−3/2∑�nt�

k=1 MkXk−1

)D−→(∫ t

0 X 2s ds,

∫ t0 Xs dMs

), hence

n3/2 (mn −1) =n−3/2∑n

k=1 Xk−1Mk

n−3∑nk=1 X2

k−1

D−→∫ 1

0 Xt dMt∫ 1

0 X 2t dt

D= N (0,σ2crit),

since∫ 1

0 X 2t dt = 1

3λ2 and

∫ 10 Xt dMt

D= N(0, 1

3λ2b2). ��

The case of unknown immigration mean has been described by Ispany et al. [15].Rahimov [22] has proved that in the non-degenerate case σ2 > 0 if the immigrationis time-dependent with regularly varying mean and variance then the limit is normalor certain functional of a time-changed Wiener process. The aymptotics of weightedCLSE has been studied in [23] using the same framework.

Acknowledgements This research was supported by the Hungarian Scientific Research Fund un-der Grant No. OTKA T-079128.

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Chapter-10

Query No. Page No. Query

AQ1 146 Please update volume id and page range for reference[13]

Critical branching processes with immigration

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