DHAHRAN 31261 SAUDI ARABIA www.kfupm.edu.sa/math/ E-mail: c-math@kfupm.edu.sa

King Fahd University of Petroleum & Minerals

DEPARTMENT OF MATHEMATICAL SCIENCES

Technical Report Series

TR 403

Dec 2008

A threshold for validity of the bootstrap in a critical branching process

I. Rahimov

A threshold for validity of the bootstrap in acritical branching process

I. Rahimov ∗

King Fahd University of Petroleum and Minerals

Abstract

In applications of branching processes, usually it is hard to obtain samplesof a large size. Therefore, a bootstrap procedure allowing inference based ona small sample size is very useful. Unfortunately, in the critical branchingprocess with a stationary immigration the standard parametric bootstrap isinvalid. In this paper we consider the process with non-stationary immigra-tion, whose mean and variance vary regularly with nonnegative exponents αand β, respectively. We prove that 1 + 2α is the threshold for the validity ofthe bootstrap in this model. If β < 1 + 2α, the standard bootstrap is validand if β > 1 + 2α it is invalid. In the case β = 1 + 2α the validity of thebootstrap depends on the slowly varying parts of the immigration mean andvariance. These results allow us to develop statistical inferences about theparameters of the process in its early stages.

MSC: primary 60J80; cecondary 62F12; 60G99.

Keywords: Branching process; Non-stationary immigration; Parametric boot-strap; Threshold; Martingale theorem; Skorokhod space.

∗Corresponding address: Department of Mathematics and Statistics, KFUPM, Box1339, Dhahran 31261, KSA, E-mail address: rahimov@kfupm.edu.sa .

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1 Introduction

We consider a discrete time branching stochastic process Z(n), n ≥ 0, Z(0) =0. It can be defined by two families of independent, nonnegative integer val-ued random variables Xni, n, i ≥ 1 and ξn, n ≥ 1 recursively as

Z(n) =

Z(n−1)∑i=1

Xni + ξn, n ≥ 1. (1.1)

Assume that Xni have a common distribution for all n and i, and familiesXni and ξn are independent. Variables Xni will be interpreted as thenumber of offspring of the ith individual in the (n− 1)th generation and ξn

is the number of immigrating individuals in the nth generation. Then Z(n)can be considered as the size of nth generation of the population.

In this interpretation, a = EXni is the mean number of offspring of asingle individual. Process Z(n) is called subcritical, critical or supercriticaldepending on a < 1, a = 1 or a > 1, respectively. The independence assump-tion of families Xni and ξn means that reproduction and immigrationprocesses are independent. However, unlike in the classical models, we donot assume that ξn, n ≥ 1, are identically distributed, i. e. the immigrationdistribution depends on the time of immigration. It is well known that as-ymptotic behavior of the process with immigration is very sensitive to anychanges of the immigration process in time.

If a sample Z(k), k = 1, ..., n is available, then the weighted conditionalleast squares estimator of the offspring mean is known to be (see [15])

an =

∑nk=1(Z(k)− α(k))∑n

k=1 Z(k − 1), α(k) = Eξk . (1.2)

In the process with a stationary immigration the maximum likelihoodestimators (MLE) for the offspring and immigration means, which were de-rived in [3] for the power series offspring and immigration distributions, arebased on the sample of pairs (Z(k), ξk), k = 1, ..., n. The MLE for the off-spring mean has the same form as an with ξk in place of α(k), and the MLEfor the immigration mean is just the average of the number of immigratingindividuals.

Sriram [17] investigated validity of the bootstrap estimator of the off-spring mean based on MLE and demonstrated that in the critical case the

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asymptotic validity of the parametric bootstrap does not hold. Similar inva-lidity of the parametric bootstrap for the first order autoregressive processwith autoregressive parameter ±1 was earlier proved in [2]. The main causeof the failure is the fact that in the critical case the MLE does not have thedesired rate of convergence (faster than n−1).

Recently, it was shown [14] that in the process with non-stationary im-migration when the immigration mean tends to infinity the conditional leastsquares estimator (CLSE) has a normal limit distribution and the rate ofconvergence of the CLSE is faster than n−1. In connection with this thefollowing question is of interest. Will the bootstrap version of the weightedCLSE of the offspring mean have the same limiting distribution as the orig-inal CLSE? In other words, will the standard parametric bootstrap be validin this non-classical model? The present paper addresses this question. Itturned out that the validity of the bootstrap depends on the relative rate ofthe immigration mean and variance. Assuming that the immigration meanand variance vary regularly with nonnegative exponents α and β, respectively,we prove that if β < 1 + 2α, the bootstrap leads to a valid approximationfor the CLSE. If β > 1 + 2α, i.e. the rate of the immigration variance isrelatively large, the conditional distribution of the bootstrap version of theCLSE has a random limit (in distribution). In the threshold case β = 1+2αthe validity depends on slowly varying parts of the mean and variance ofthe immigration. To prove these results, first we obtain approximation the-orems for an array of nearly critical processes, which are also of independentinterest.

It follows from the above discussion that the question on criticality ofthe process is crucial for applications. To answer this question, one may testhypothesis H0 : a = 1 against one of a 6= 1, a > 1 or a < 1. Our results allowto develop rejection regions for these hypotheses based on bootstrap pivots(equation (2.6)). Sampling distributions of the bootstrap pivots can easilybe generated based on a single sample Z(k), k = 1, ..., n with relativelysmall generation number n. Thus, the bootstrap procedure is very useful inapplications of branching processes, where it is hard to obtain samples of alarge size. It allows statistical inferences in early stages of the process. This isimportant, for example, in epidemic models, which can be approximated bybranching processes when initial number of susceptible individuals is large.

Investigation of the problems related to the bootstrap methods and theirapplications has been an active area of the research since its introduction byEfron [9]. As a result a large number of papers and monographs have been

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published. We note monographs [8] and [10] and the most recent review arti-cles [7] and [13] as important sources of the literature on bootstrap methods.As it was mentioned before, invalidity of the bootstrap for the critical processwith a stationary immigration was shown in [17]. In [6] a modification of thestandard bootstrap procedure was proposed, which eliminated the invalidityin the critical case. The second-order correctness of the bootstrap for a stu-dentized version of MLE in subcritical case proved in [18].

The paper is organized as follows. In Section 2 of the paper, we describe theparametric bootstrap and state the main results. Necessary limit theoremsfor an array of branching processes and their consequences related to CLSEwill be derived in Section 3. The proofs of main theorems are given inSection 4. Appendix contains the proofs of limit theorems for the array ofnearly critical processes.

2 Main results on the bootstrap

The process with time-dependent immigration is given by the offspringdistribution of Xki, k, i ≥ 1, and by the family of distributions of the numberof immigrating individuals ξk, k ≥ 1. We assume that the offspring distribu-tion has the probability mass function

pj(θ) = PXki = j, j = 0, 1, ..., (2.1)

depending on parameter θ, where θ ∈ Θ ⊆ R. Then a = EθXki = f(θ)for some function f . We assume throughout the paper that f is one-to-onemapping of Θ to [0,∞) and is a homeomorphism between its domain andrange. It is known that these assumptions are satisfied, for example, by thedistributions of the power series family [6].

About the number of immigrating individuals, we assume that ξk followsa known distribution with the probability mass function

qj(k) = Pξk = j, j = 0, 1, ..., (2.2)

for any k ≥ 1.

Throughout the paper ”D→ ”, ”

d→ ” and ”P→ ” will denote convergence

of random functions in Skorokhod topology and convergence of random vari-

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ables in distribution and in probability, respectively, and also Xd= Y de-

notes equality of distributions. We assume that b = V arXni < ∞ andα(k) = Eξk, β(k) = V arξk are finite for any k ≥ 1 and are regularlyvarying sequences of nonnegative exponents α and β, respectively. ThenA(n) = EZ(n) and B2(n) = V arZ(n) are finite for each n ≥ 0 and a = 1.To provide the asymptotic distribution of an defined in (1.2), we assume thatthere exists c ∈ [0,∞] such that

limn→∞

β(n)

nα(n)= c (2.3)

and denote for any ε > 0

δn(ε) =1

B2(n)

n∑

k=1

E[(ξk − α(k))2; |ξk − α(k)| > εB(n)].

As it was proved in [15], if a = 1, b ∈ (0,∞), α(n) →∞, β(n) = o(nα2(n)),condition (2.3) is satisfied and δn(ε) → 0 as n →∞ for each ε > 0 , then asn →∞

nA(n)

B(n)(an − a)

d→ (2 + α)N (0, 1). (2.4)

Furthermore, under the above conditions, A(n)/B(n) → ∞ as n → ∞ andwhen c = 0 the condition δn(ε) → 0 is satisfied automatically. More detaileddiscussion and examples can be seen in [15].

We now describe the bootstrap procedure to approximate the samplingdistribution of the pivot

Vn =nA(n)

B(n)(an − a).

Given a sample Xn = Z(k), k = 1, ..., n of population sizes, we esti-mate the offspring mean a by the weighted CLSE an. Obtain the esti-mate of the parameter θ as θn = f−1(an) from equation a = f(θ). Re-place θ in the probability distribution (2.1) by its estimate. Given Xn, let

X∗(n)ki , k, i ≥ 1 be a family of i.i.d. random variables with the probability

mass function pj(θn), j = 0, 1, .... Now we obtain the bootstrap sampleX ∗

n = Z∗(n)(k), k = 1, ..., n recursively from

Z∗(n)(k) =

Z∗(n)(k−1)∑i=1

X∗(n)ki + ξk, n, k ≥ 1, (2.5)

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with Z∗(n)(0) = 0, where ξk, k ≥ 1, are independent random variables withthe probability mass functions qj(k), j = 0, 1, .... Then, we define thebootstrap analogue of the pivot Vn by

V ∗n =

nA(n)

B(n)(a∗n − an), (2.6)

where a∗n is the weighted CLSE based on X ∗n , i.e.

a∗n =

∑nk=1(Z

∗(n)(k)− α(k))∑nk=1 Z∗(n)(k − 1)

. (2.7)

To state our main result, we need the following conditions be satisfied.

A1. a = 1 and moments Eθ[(Xki)2] and Eθ[(Xki)

2+l] are continuous functionsof θ for some l > 0.A2. δn(ε) → 0 as n →∞ for each ε > 0.A3. α(n) →∞, β(n) = o(nα2(n)) as n →∞.

Theorem 2.1. If conditions A1-A3 and (2.3) are satisfied, then

supx|PV ∗

n ≤ x|Xn − Φ(2 + α, x)| P→ 0 (2.8)

as n →∞, where Φ(σ, x) is the normal distribution with mean zero and vari-ance σ2.

Remarks. 2.1. Due to (2.4) convergence (2.8) implies the validity of thestandard bootstrap procedure to approximate the sampling distribution ofVn, i.e. as n →∞

supx|PV ∗

n ≤ x|Xn − PVn ≤ x| P→ 0.

2.2. As it was mentioned before, in the case c = 0 condition A2 is automati-cally satisfied. In the case c > 0 the condition is equivalent to the Lindebergcondition for the family ξn, n ≥ 1 of the number of immigrating individuals.

Example 2.1. Let ξk, k ≥ 1, be Poisson with the mean α(k) such thatα(k) → ∞, k → ∞, and regularly varies with exponent α. In this caseβ(n) = o(nα(n)) as n → ∞ and condition A3 is satisfied. Moreover, we

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realize that c = 0 in (2.3), which implies that condition A2 is also fulfilled.Thus we have the following result.

Corollary 2.1. If ξk, k ≥ 1, are Poisson with mean α(k) →∞, k →∞, and(α(k))∞k=1 is regularly varying with exponent α and condition A1 is satisfied,then (2.8) holds.

Now we consider the case of the fast immigration variance. In other words,we assume that there exists d ∈ [0,∞) such that

limn→∞

nα2(n)

β(n)= d. (2.9)

It is known that in this case the weighted CLSE is not asymptotically normal.More precisely, if a = 1, b ∈ (0,∞), α(n) → ∞, condition (2.9) is satisfiedand δn(ε) → 0 as n →∞ for each ε > 0 , then as n →∞

n(an − a)d→ W0 =:

W (1)∫ 1

0W (t1+β)dt + γ

, (2.10)

where W (t) is the standard Wiener process and γ = ((α+1)(α+2))−1√

d(1 + β)(see [15], Theorem 3.2).

We denote by Wn = n(an − a) the pivot corresponding to this result andby W ∗

n = n(a∗n − an) the bootstrap pivot based on the bootstrap sample X ∗n .

The next theorem shows that in the case of fast immigration variance theparametric bootstrap is invalid. To formulate corresponding theorem, weneed several new notations. Let ∇β(t) = ∇β(c0, t) = µβ(2c0, t), where

µα(t) = µα(c0, t) =

∫ t

0

uαe(t−u)c0du, γ0 = γ0(c0) =

(d

∇β(1)

)1/2 ∫ 1

0

µα(u)du,

ψ(t) = ψ(c0, t) =t1+β

(1 + β)∇β(1), Λ(c0, t) = W (ψ(t))+c0

∫ t

0

ec0(t−u)W (ψ(u))du.

Further, introduce ratio

ν(c0) =W (ψ(c0, 1))∫ 1

0Λ(c0, t)dt + γ0(c0)

(2.11)

and its cumulative distribution function F (c0, x) = Pν(c0) ≤ x.

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Theorem 2.2. If conditions A1, A2 and (2.9) are satisfied and α(n) → ∞as n →∞, then

PW ∗n ≤ x|Xn d→ F (W0, x) (2.12)

as n →∞ for each x ∈ R.

Remark 2.3. Theorem 2.2 shows that, when the immigration varianceis large enough comparatively the mean, the conditional limit distributionof the bootstrap pivot W ∗

n does not coincide with the limit distribution ofWn = n(an − a) for a = 1. In other words, as in the case of stationaryimmigration, the standard bootstrap least squares estimate of the offspringmean is asymptotically invalid. It is not surprising, if we recall that even thebootstrap version of the sample mean is invalid in the infinite variance case[1]. In this case one may develop a modified version of the bootstrap as in[6] (see Section 5 for details).

In order to prove the main theorems, we obtain a series of results for thearray of the branching processes in a more general set up, which are ofindependent interest as well. The scheme of the proofs is as following. Sincethe bootstrap sample X ∗

n is based on the sequence of branching processes(2.5), we first investigate the array of processes under suitable assumptionsof the nearly criticality. In the second step, we derive limit distributions forthe CLSE of the offspring mean in the sequence of nearly critical processes.In the third step, we show that conditions of the limit theorems for the CLSEare fulfilled by the bootstrap pivots V ∗

n and W ∗n .

3 Approximation of the sequence of processes

We now consider a sequence of branching processes defined as following.Let X(n)

ki , k, i ≥ 1 and ξ(n)k , k ≥ 1 be two families of independent, non-

negative and integer valued random variables for each n ∈ N. The sequenceof branching processes (Z(n)(k), k ≥ 0)n≥1 with Z(n)(0) = 0, n ≥ 1, is definedrecursively as

Z(n)(k) =

Z(n)(k−1)∑i=1

X(n)ki + ξ

(n)k , n ≥ 1. (3.1)

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Assume that X(n)ki have a common distribution for all k and i, and families

X(n)ki and ξ(n)

k are independent. As before, the variables X(n)ki will be

interpreted as the number of offspring of the ith individual in the (k − 1)th

generation and ξ(n)k is the number of immigrating individuals in the kth gen-

eration. Then Z(n)(k) can be considered as the size of population of kthgeneration in nth process.

In this scheme, a(n) = EX(n)ki is the criticality parameter of the nth

process. The process Z(n)(k), k ≥ 0 for each fixed n is called subcritical,critical or supercritical depending on a(n) < 1, a(n) = 1 or a(n) > 1, respec-tively. The sequence of branching processes (3.1) is said to be nearly criticalif a(n) → 1 as n → 1. In this section we investigate asymptotic behavior ofa sum of normalized martingale differences generated by the nearly criticalsequence of branching processes with non-stationary immigration, under theassumption that sequences of the immigration means and variances can beapproximated by regularly varying sequences with non-negative exponents.We prove that when the immigration mean tends to infinity depending onthe time of immigration, the ”broken-line” process can be approximated inSkorokhod topology by a time-changed Wiener process.

If a sequence (f(n))∞n=1 is regularly varying with exponent ρ, we will

write (f(n))∞n=1 ∈ Rρ. We assume that a(n) = EX(n)ij and b(n) = V arX

(n)ij

are finite for each n ≥ 1 and α(n, i) = Eξ(n)i < ∞, β(n, i) = V arξ

(n)i < ∞

for all n, i ≥ 1. Then An(i) = EZ(n)(i) and B2n(i) = V arZ(n)(i) are finite for

each n ≥ 1, 1 ≤ i ≤ n, and by a standard technique we find that

An(k) =k∑

i=1

α(n, i)ak−i(n), B2n(k) = ∆2

n(k) + σ2n(k), (3.2)

where

∆2n(k) =

k∑i=1

α(n, i)V ar(X(n)(k − i)), σ2n(k) =

k∑i=1

β(n, i)a2(k−i)(n),

V ar(X(n)(i)) =b(n)

1− a(n)ai−1(n)(1− ai(n)).

We also denote by =(n)(k) for each n ≥ 1 the σ-algebra containingall the history of the nth process up to kth generation, i.e. it is gener-ated by Z(n)(0) , Z(n)(1) , ..., Z(n)(k) and put M (n)(k) = Z(n)(k) −

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E[Z(n)(k)|=(n)(k)]. In our proofs we need approximation results for the fol-lowing processes

Zn(t) =Z(n)([nt])

An(n), Yn(t) =

1

Bn(n)

[nt]∑

k=1

M (n)(k), t ∈ R+.

In the approximation we assume the following conditions are satisfied.

C1. There are sequences (α(i))∞i=1 ∈ Rα and (β(i))∞i=1 ∈ Rβ with α, β ≥ 0,such that, as n →∞ for each s ∈ R+,

max1≤k≤ns

|α(n, k)− α(k)| = o(α(n)), max1≤k≤ns

|β(n, k)− β(k)| = o(β(n)). (3.3)

C2. a(n) = 1 + n−1c0 + o(n−1) as n →∞ for some c0 ∈ R.C3. b(n) = o(α(n)) as n →∞.

Remark 3.1. The condition C2 is the same is in the study of the ar-ray of time-homogeneous processes. C3 includes the cases, where the off-spring variance tends to infinity or to zero. The first part of conditionC1, related to the immigration mean, is satisfied when α(n) → ∞, if justlimn→∞ max1≤k≤ns |α(n, k) − α(k)| < ∞. In general, C1 is satisfied, for ex-ample, if there are εi(n) → 0 as n → ∞, i = 1, 2, such that α(n, k) =α(k)(1 + ε1(n)) and β(n, k) = β(k)(1 + ε2(n)).

We obtain three different approximations for Yn(t), depending on which ofthe following conditions holds.

C4. α(n) →∞, β(n) = o(nα(n)b(n)) as n →∞ and lim infn→∞ b(n) > 0.C5. α(n) →∞ and nα(n)b(n) = o(β(n)) as n →∞.C6. α(n) →∞ and β(n) ∼ cnα(n)b(n) as n →∞, where c ∈ (0,∞).

The following functions appear in the time change of approximating processes:

µα(t) =

∫ t

0

uαe(t−u)c0du, να(t) =

∫ t

0

uαe(t−u)c0(1− e(t−u)c0)du. (3.4)

In particular, it is useful to note that µα(t) = tα+1/(α + 1) when c0 = 0, andlimc0→0 να(t)/c0 = tα+2/(α + 1)(α + 2).

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We denote δ(1)n (ε) = E[(X

(n)ki −a(n))2χ(|X(n)

ki −a(n)| > εBn(n))], where χ(A)stands for the indicator of event A. We denote

ϕ(t) =

c0να(1)

∫ t

0µα(u)du, if c0 6= 0,

t2+α, if c0 = 0.

Theorem 3.1. If conditions C1-C4 are satisfied and δ(1)n (ε) → 0 as n →∞,

then YnD→ Y (1) as n → ∞ weakly in Skorokhod space D(R+,R), where

Y(1)(t) = W (ϕ(t)) and (W (t), t ∈ R+) is a standard Brownian motion.

Remark 3.2. We note that the Lindeberg-type condition for the familyX(n)

ki , k, i ≥ 1 is needed even in time-homogenous models (see [11], [17]).

If E(X(n)ki )2+l < ∞ for all n ∈ N and some l ∈ R+, then

δ(1)n (ε) ≤ 1

εlBln(n)

E|X(n)ki − a(n)|2+l.

Since when C4 is satisfied, B2n(n) ∼ Kn2α(n)b(n) as n → ∞, where K is a

positive constant, the Lindeberg-type condition is satisfied, for example, ifE|X(n)

ki − a(n)|3 = o(n√

α(n)b(n)) as n →∞.

To state next theorem, we need the Lindeberg condition for the family ofimmigrating individuals ξ(n)

k , k ≥ 1, which is given by the following ratio

δ(2)n (ε) =

1

σ20(n)

n∑

k=1

E[(ξ(n)k )2χ(|ξ(n)

k | > εσ0(n))], σ20(n) =

n∑

k=1

β(n, k),

where ξ(n)k = ξ

(n)k − α(n, k).

Theorem 3.2. If conditions C1-C3 and C5 are satisfied and δ(2)n (ε) → 0 as

n → ∞, then YnD→ Y (2) as n → ∞ weakly in Skorokhod space D(R+,R),

where Y(2)(t) = W (ψ(t)),

ψ(t) =t1+β

(1 + β)∇β(1).

and ∇β(t) is defined right before Theorem 2.2.

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Theorem 3.3. Let conditions C1-C3 and C6 be satisfied. If δ(i)n (ε) → 0 as

n → ∞ for each ε > 0, i = 1, 2, and lim infn→∞ b(n) > 0, then YnD→ Y (3)

as n → ∞ weakly in Skorokhod space D(R+,R), where Y(3)(t) = W (ω(t))and

ω(t) =1

K(c0, c)

∫ t

0

(µα(u) + cuβ)du,

K(c0, c) = να(1)/c0+c∇β(1) for c0 6= 0 and K(0, c) = 1/(1+α)(2+α)+cµβ(1).

Remark 3.3. It is not difficult to see that for c0 = 0, the limiting processhas a simpler time change, namely

ω(t) = (1

(1 + α)(2 + α)+

c

1 + β)−1[

t2+α

(1 + α)(2 + α)+ c

t1+β

1 + β].

Furthermore, if in addition b(n) → b as n → ∞ with b ∈ (0,∞), thenω(t) = tα+2 = tβ+1, i.e. the same time change as in the functional limittheorem for the critical process [14].

The proofs of Theorems 3.1-3.3 are provided in the Appendix. We also needthe following theorem on a deterministic approximation of the process Zn(t),which is proved in [16].

Theorem 3.4. Let conditions C1-C3 be satisfied. If α(n) →∞ and β(n) =

o(nα2(n)) as n → ∞, then ZnD→ πα as n → ∞ weakly in Skorokhod space

D(R+,R), where πα(t) = µα(t)/µα(1), t ∈ R+.

We now derive two important results from Theorems 3.1-3.5, which arerelated to the CLSE of the offspring mean in the nearly critical sequence ofbranching processes. By a standard technique we obtain that the CLSE ofthe offspring mean a(n) of nth process in (3.1) is

An =

∑nk=1(Z

(n)(k)− α(n, k))∑nk=1 Z(n)(k − 1)

. (3.5)

We denote

W (1)n =

nAn(n)

Bn(n)(An − a(n)), W (2)

n = n(An − a(n)). (3.6)

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Theorem 3.5. Let conditions C1-C3 be satisfied and α(n) → ∞, β(n) =o(nα2(n)) as n →∞.

a) If C4 holds and δ(1)n (ε) → 0 as n →∞ for any ε > 0, then

W (1)n

d→ W (ϕ(1))∫ 1

0πα(u)du

.

b) If C5 holds and δ(2)n (ε) → 0 as n →∞ for any ε > 0, then

W (1)n

d→ W (ψ(1))∫ 1

0πα(u)du

.

c) If C6 holds, δ(i)n (ε) → 0, i = 1, 2, as n → ∞ for any ε > 0 and

lim infn→∞ b(n) > 0, then

W (1)n

d→ W (ω(1))∫ 1

0πα(u)du

.

Proof. We consider the following equality:

An − a(n) =

∑nk=1 M (n)(k)∑n

k=1 Z(n)(k − 1)=:

D(n)

Q(n). (3.7)

First we obtain the asymptotic behavior of Q(n). We use the followingrepresentation

1

nAn(n)

n∑

k=0

Z(n)(k) =n∑

k=1

∫ k/n

(k−1)/n

Zn(t)dt. (3.8)

Now we consider functionals Ψn : D(R+,R) 7→ R, defined for any x ∈D(R+,R) and n ≥ 1 as

Ψn(x) =n∑

k=1

∫ k/n

(k−1)/n

x(t)dt. (3.9)

It is obvious that for all x, xn ∈ D(R+,R) such that ‖xn−x‖∞ → 0, we have|Ψn(xn)−Ψ(x)| → 0 as n →∞, where

Ψ(x) =

∫ 1

0

x(t)dt.

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Due to Theorem 3.4 we have ZnD→ πα as n →∞ weakly in Skorokhod space

D(R+,R). Since πα(t) is continuous, this implies that ‖Zn − πα‖∞ → 0as n → ∞. Therefore, due to the extended continuous mapping theo-

rem ([4], Theorem 5.5), we have Ψn(Zn)d→ Ψ(πα) as n → ∞, where

Ψ(πα) =∫ 1

0πα(t)dt. Thus, when conditions C1-C3 are satisfied and α(n) →

∞, β(n) = o(nα2(n)) as n →∞, we have

Q(n)

nAn(n)

P→∫ 1

0

πα(u)du. (3.10)

Now we consider D(n). It follows from Theorem 3.1 that when conditions

C1-C4 are satisfied and δ(1)n (ε) → 0 as n →∞ for any ε > 0 as n →∞

D(n)

Bn(n)

d→ W (ϕ(1)). (3.11)

The proof of Part (a) of the theorem follows from (3.7), (3.10) and (3.11)due to Slutsky’s theorem.

Using Theorems 3.2 and 3.3, by similar arguments we obtain parts (b)and (c) of Theorem 3.5.

The following consequence of the above theorem is vital in the proof ofthe main results.

Corollary 3.1. If conditions of Theorem 3.5 are satisfied and c0 = 0, then

W(1)n

d→ (2 + α)N (0, 1) as n →∞ in all cases (a), (b) and (c).

The proof of the corollary is straightforward and uses the fact that µα(u) =u1+α/(1 + α) when c0 = 0.

The second approximation, related to the pivot W(2)n , holds under the

assumption that there exists d ∈ [0,∞) such that the condition (2.9) holds.

Theorem 3.6. Let conditions C1, C2 and (2.9) be satisfied. If α(n) →∞, b(n) → b ∈ (0,∞) and δ

(2)n (ε) → 0 as n →∞ for any ε > 0, then

W (2)n

d→ ν(c0), (3.12)

where ν(c0) is defined in (2.11).

14

Proof. We consider relation (3.7) in the proof of Theorem 3.5. It is obviousthat conditions C3 and C5 of Theorem 3.2 are satisfied when (2.9) holdsand b(n) → b ∈ (0,∞) as n → ∞. Therefore, taking into account thatD(n)/B(n) = Yn(1), we obtain that as n →∞

D(n)

Bn(n)=

d→ W (ψ(1)). (3.13)

Now we evaluate Q(n). When the condition β(n) = o(nα2(n)) as n → ∞is not valid we can not use the deterministic approximation for Zn(t) givenin Theorem 3.4. Therefore, first we express the denominator in terms of theprocess

Yn(t) =Z(n)([nt])− A(n)([nt])

Bn(n).

Namely, we consider the following equality

Q(n)

nBn(n)= S1(n) + S2(n), (3.14)

where

S1(n) =n∑

k=1

∫ k/n

(k−1)/n

Yn(t)dt, S2(n) =1

nBn(n)

n∑

k=1

An(k − 1).

To evaluate S2(n), we use Lemma 2 and Part (c) of Lemma 3, which areproved in the Appendix. Since due to condition (2.9) B2

n(n) ∼ nβ(n)∇β(1)as n →∞, we easily obtain that

limn→∞

S2(n) = γ0, (3.15)

where γ0 = γ0(c0) is defined just before Theorem 2.2. In order to usethe continuous mapping theorem, we need to express S1(n) in terms of theprocess Yn(t). We obtain using (3.1) that Z(n)(k) − EZ(n)(k) = M (n)(k) +a(n)(Z(n)(k − 1)− EZ(n)(k − 1)) for any n, k ≥ 1. Thus

Z(n)(k)− EZ(n)(k) =k∑

j=1

ak−j(n)M (n)(j).

15

From the last equality we easily obtain

Yn(t) = Yn(t) +1

Bn(n)

[nt]∑j=1

(a[nt]−j(n)− 1)M (n)(j). (3.16)

Rearranging the sum on the right side of (3.16), we have

1

Bn(n)

[nt]∑j=1

(a[nt]−j(n)− 1)M (n)(j) =a(n)− 1

Bn(n)

[nt]∑j=1

[nt]∑i=j+1

a[nt]−i(n)M (n)(j) =

= (a(n)− 1)

[nt]∑i=2

a[nt]−i(n)Yn(i− 1

n).

Hence, we obtain the following representation

S1(n) =n∑

k=1

∫ k/n

(k−1)/n

Yn(t) + c0(n)

1

n

[nt]∑i=2

a[nt]−i(n)Yn(i− 1

n)

dt, (3.17)

where c0(n) = n(a(n) − 1). Now we consider sequence of functionals Φn :D(R+,R) 7→ R, n ≥ 1, which are defined for any x ∈ D(R+,R) as

Φn(x) =x(1)

Ωn(x) + γ0

,

where functionals Ωn : D(R+,R) 7→ R, n ≥ 1 are defined by

Ωn(x) =n∑

k=1

∫ k/n

(k−1)/n

x(t) + c0(n)

1

n

[nt]∑i=2

a[nt]−i(n)x(i− 1

n)

dt. (3.18)

It is not difficult to see that for any sequence xn ∈ D(R+,R), n ≥ 1, suchthat ‖xn − x‖∞ → 0 as n →∞ with x ∈ D(R+,R), we have Φn(xn) → Φ(x)as n →∞, where

Φ(x) = x(1)

(∫ 1

0

x(t)dt + c0

∫ 1

0

∫ t

0

e(t−u)c0x(u)dudt + γ0

)−1

. (3.19)

It follows from Theorem 3.2 that YnD→ Y (2) as n →∞ weakly in Skorokhod

space D(R+,R), where Y (2)(t) = W (ψ(t)). Since Y(2)(t) is continuous, this

16

implies that ‖Yn −Y(2)‖∞ → 0 as n →∞. Appealing again to the extended

continuous mapping theorem ([4], Theorem 5.5), we conclude that Φn(Yn)d→

Φ(Y(2)) as n →∞. To obtain the proof of Theorem 3.6, we rewrite (3.7) as

n(An − a(n)) =Φn(Yn)

1 + Υn(Yn),

where Υn(Yn) = (S2(n)− γ0)/(Ωn(Yn) + γ0). Since Υn(Yn)P→ 0 as n →∞,

the assertion of Theorem 3.6 now follows from Slutsky’s theorem.

Corollary 3.2. If conditions of Theorem 3.6 are satisfied and c0 = 0, then

W (2)n

d→ W (1)∫ 1

0W (t1+β)dt + γ

(3.20)

as n →∞, where γ = ((α + 1)(α + 2))−1√

d(1 + β).

The proof of the corollary is straightforward.

4 Proofs of the main results

Proof of Theorem 2.1. It is easy to see that given the sample Xn =Z(k), k = 1, ..., n, the bootstrap process Z∗(n)(k), k ≥ 0 is the sequenceof branching processes defined in (3.1), where

a(n) = an, b(n) = V ar(X∗(n)ki ),

and α(n, k) = α(k) and β(n, k) = β(k) are the mean and variance of theprobability distribution qj(k), j = 0, 1, .... Therefore, if conditions ofTheorem 3.5 are satisfied by the bootstrap process (in probability) and c0 =0, then

supx|Hn(an, x)− Φ(2 + α, x)| P→ 0, (4.1)

where Hn(an, x) = PV ∗n ≤ x|Xn. Thus, we need to show that the con-

ditions of Theorem 3.5 are satisfied. Conditions C1 are fulfilled trivially.

It follows from (2.4) that n(an − 1)P→ 0 as n → ∞, i.e. the condition

C2 is also satisfied in probability with c0 = 0. Since anP→ 1 as n → ∞,

17

θn = f−1(an)P→ f−1(1) = θ. Since b(n) = ϕ1(θ), where ϕ1 is a continuous

function, we have V ar(X∗(n)ki |Xn) = ϕ1(θn)

P→ ϕ1(θ) = b as n → ∞, i.e.condition C3 is satisfied.

It follows from Remark 3.2 that if for some l > 0

(1/Bln(n))E[|X∗(n)

ki − an|2+l|Xn]P→ 0 (4.2)

as n →∞, then

δ∗(1)n (ε) =:

1

B2n(n)

E[(X∗(n)ki − an)2χ(|X∗(n)

ki − an| > εBn(n))|Xn]P→ 0

as n →∞ for each ε > 0. We obtain from condition A1 that E[(X∗(n)ki )2+l|Xn]

P→E[(Xki)

2+l] as n →∞, which implies (4.2). Thus, all conditions of Theorem3.5 are satisfied and c0 = 0. It follows from Lemmas 1 and 2 in the Appendixthat An(n) ∼ A(n) and B2

n(n) ∼ B2(n) as n → ∞ when c0 = 0 and theassertion of (2.8) follows from Corollary 3.1. Theorem 2.1 is proved.

Proof of Theorem 2.2. As in [17], we use quite standard technique basedon Skorokhod’s theorem (see [5], Theorem 29.6). We have from (2.10) that

n(an−1)d→ W0 as n →∞. Therefore, due to the Skorokhod’s theorem there

exists a sequence a′n, n ≥ 1 of random variables and a random variable

W ′0 on a common probability space (Ω′,F , Q) such that a′n

d= an for all

n ≥ 1, W ′0

d= W0 and n(a′n(ω′)− 1) → W ′

0 as n →∞ for each ω′ ∈ Ω′.For any ω′ ∈ Ω′ we estimate unknown θ by θ′n(ω′) = f−1(a′n(ω′)). Then

we obtain the bootstrap distribution pj(θ′n), j ≥ 0 substituting θ by θ′n(ω′).

Let now X ′(n)ki , k, i ≥ 1 be a family of i.i.d. random variables such that

PX ′(n)ki = j = pj(θ

′n)

for each ω′ ∈ Ω′ and n ≥ 1 and ξk, k ≥ 1 be a sequence of random variableswith the probability distributions qj(k), j ≥ 0. A new bootstrap sampleX ′

n = Z ′(n)(k), k = 1, ..., n will be obtained recursively from the relation

Z ′(n)(k) =

Z′(n)(k−1)∑i=1

X′(n)ki + ξk, k = 1, 2, ... (4.3)

18

for each ω′ ∈ Ω′, n ≥ 1 with Z ′(n)(0) = 0. We define new pivot W ′n =

n(an − a′n(ω′)) for each ω′ ∈ Ω′, where

an =

∑nk=1(Z

′(n)(k)− α(k))∑nk=1 Z ′(n)(k − 1)

. (4.4)

If we denote Fn(θ, x) = PWn ≤ x, we realize that

PW ∗n ≤ x|an = Fn(θn, x), PW ′

n ≤ x|a′n = Fn(θ′n, x),

where θn = f−1(an) and θ′n = f−1(a′n). For each ω′ ∈ Ω′ we apply Theo-rem 3.6 to W ′

n. Under the assumptions of Theorem 2.2 conditions C1 and

δ(2)n (ε) → 0 as n → ∞ for any ε > 0 are trivially satisfied. Condition C2

is also fulfilled for each ω′ ∈ Ω′ with c0 = W ′0. Due to our assumptions

on the moments of the offspring distribution we can write b(n) = ϕ1(θ),where ϕ1 is a continuous function, as in the proof of Theorem 2.1. There-fore, V ar(X

′(n)ki |θ′n) = ϕ1(θ

′n) → ϕ1(θ) = b as n →∞ for each ω′ ∈ Ω′. Thus

we obtain from Theorem 3.6 that Fn(θ′n, x) → F (W ′0, x) as n → ∞ for each

ω′ ∈ Ω′ and x ∈ R. The assertion of Theorem 2.1 now follows from this dueto Fn(θn, x)

d= Fn(θ′n, x) and F (W0, x)

d= F (W ′

0, x). The theorem is proved.

5 Conclusions

According to Theorem 2.2, when nα2(n) = o(β(n)) as n → ∞ the boot-strap version of CLSE is invalid. The cause of the failure is the same as inthe case of stationary immigration [17], namely, in this case the estimatoran does not have the desired rate of convergence to a = 1. As in [6], onemay consider a modified version of the standard bootstrap procedure. Theidea behind the modification is using in the initial estimator of the offspringmean an adaptive shrinkage towards a = 1.

If a sample of pairs (Z(k), ξk), k = 1, ..., n is available, then a naturalestimator of the offspring mean is

an =

∑nk=1(Z(k)− ξk)∑n

k=1 Z(k − 1).

The following questions related to this estimator is of interest. How muchimprovement in a sense of the rate of convergence we will get because of

19

additional observations of the number of immigrating individuals? Will thestandard parametric bootstrap procedure be valid for an in the case of largeimmigration variance? Since

an − a =

∑nk=1

∑Z(k−1)j=1 (Xkj − a)∑n

k=1 Z(k − 1),

one can easily derive asymptotic distributions for the pivot, correspondingto an from a martingale central limit theorem. By the arguments as in theproof of Proposition 4.1 in [15], it is possible to prove that an is a stronglyconsistent estimator of a.

The estimation problems and a justification of the validity of the boot-strap for subcritical and supercritical processes with non-stationary immi-gration are also open. In order to derive the asymptotic distributions foran estimator of the offspring mean, one needs to establish functional limittheorems in these cases. Further, as in the classical models, one may obtainresults for the estimator without any assumption of the criticality.

6 Appendix

In the proofs we use several preliminary lemmas. We start with a simplebut useful result related to regularly varying sequences.

Lemma 1. If (C(n))∞n=1 ∈ Rρ and a(n) satisfies condition C2, then for anyρ ∈ [0,∞) and θ ∈ R

1

nC(n)

[ns]∑

k=1

akθ(n)C(k) →∫ s

0

tρetθc0dt (6.1)

as n →∞ uniformly in s ∈ [0, T ] for each fixed T > 0.

Proof. It follows from condition C2 that

ans(n) → ec0s (6.2)

as n →∞ uniformly in s ∈ [0, T ]. Since (C(n))∞n=1 ∈ Rρ, (6.2) implies that

1

n

[ns]∑

k=1

C(k)

C(n)akθ(n)− 1

n

[ns]∑

k=1

(k

n)ρek/nθc0 → 0

20

as n → ∞ uniformly in s ∈ [0, T ]. Now the second sum tends as n → ∞ tothe integral ∫ s

0

tρetθc0dt,

which is a continuous function of s. Hence, the convergence is uniform ins ∈ [0, T ]. The Lemma is proved.

The next result is related to the asymptotic behavior of the mean andthe variance of the process.

Lemma 2. If conditions C1 and C2 are satisfied, then uniformly in s ∈ [0, T ]for each fixed T > 0

a) limn→∞

An([ns])

nα(n)= µα(s), lim

n→∞σ2

n([ns])

nβ(n)= ∇β(s),

b) limn→∞

∆2n([ns])

n2α(n)b(n)=

(1/c0)να(s), if c0 6= 0,

sα+2/(α + 1)(α + 2), if c0 = 0.

Proof. To prove the first relation in Part (a), we consider

An([ns]) =

[ns]∑

k=1

α(k)a[ns]−k(n) +

[ns]∑

k=1

(α(n, k)− α(k))a[ns]−k(n). (6.3)

Applying Lemma 1, we easily obtain that the first term on the right side of(6.3), divided by nα(n), as n → ∞ tends to µα(s) uniformly in s ∈ [0, T ].The second term is dominated by

max1≤k≤nT

|α(n, k)− α(k)|[ns]∑

k=1

a[ns]−k(n).

Due to Lemma 1, the sum in this expression, divided by n, tends to∫ s

0e(1−u)c0du

as n → ∞ uniformly in s ∈ [0, T ]. Therefore, taking into account C1, weobtain the assertion.

The proofs of the remaining claims are similar and, therefore, are omitted.

Lemma 3. If conditions C1 and C2 are satisfied, then for any θ ∈ R, s ∈ R+

a) limn→∞

1

n3α(n)b(n)

[ns]∑i=1

aθi(n)∆2n(i) =

(1/c0)∫ s

0euθc0να(u)du, if c0 6= 0,

sα+3/(α + 1)(α + 2)(α + 3), if c0 = 0,

21

b) limn→∞

1

n2β(n)

[ns]∑i=1

aθi(n)σ2n(i) =

∫ s

0

euθc0∇β(u)du,

c) limn→∞

1

n2α(n)

[ns]∑i=1

aθi(n)An(i) =

∫ s

0

euθc0µα(u)du,

d) limn→∞

1

n2α(n)β(n)

[ns]∑i=1

aθi(n)β(i)An(i) =

∫ s

0

euθc0µα(u)uβdu.

Proof. We prove Part (a). Let c0 6= 0. In this case due to Part (b) ofLemma 2 we have

1

n

[ns]∑i=1

aθi(n)∆2

n(i)

n2α(n)b(n)− 1

nc0

[ns]∑i=1

e(i/n)θc0να(i

n) → 0

as n →∞. Since euθc0να(u) is bounded in u ∈ [0, s] for each fixed s,

limn→∞

1

n

[ns]∑i=1

e(i/n)θc0να(i

n) =

∫ s

0

euθc0να(u)du. (6.4)

In the case c0 = 0 we have∫ s

0(uα+2/(α + 1)(α + 2))du on the right side

of (6.4). The proofs of Parts (b), (c) and (d) are similar. We just use thesecond relation of Part (a) and Part (b) of the Lemma 2. Lemma 3 is proved.

Lemma 4. Let X(n)ki , k, j, n ≥ 1, be random variables from (3.1), X

(n)ki =

X(n)ki − a(n) and Tn(k) =

∑Z(n)(k−1)i=1 X

(n)ki . Then

a)

E[(Tn(k))2|=(n)(k − 1)] = b(n)Z(n)(k − 1), (6.5)

b)

E[(Σ′X(n)ki X

(n)kj )2|=(n)(k − 1)] = 2b2(n)Z(n)(k − 1)(Z(n)(k − 1)− 1), (6.6)

where∑′ denotes summation for i, j = 1, 2, ..., Z(n)(k − 1) such that i 6= j.

Proof. The proof of identities (6.5) and (6.6) follows directly from inde-

pendence of random variables X(n)ki and X

(n)kj , j 6= i, and properties of the

22

conditional expectations.

Lemma 5. For the sum Tn(k) defined in Lemma 4 and any θ > 0

E[(Tn(k))2χ(|Tn(k)| > θ)|=(n)(k − 1)] ≤ I1 + I2, (6.7)

whereI1 = Z(n)(k − 1)E[(X

(n)ki )2χ(|X(n)

ki | > θ/2)],

I2 =4b2(n)

θ2(Z(n)(k − 1))2 +

√2b3(n)

θ(Z(n)(k − 1))3/2.

Proof. Using simple inequality

χ(|ξ + η| > ε) ≤ χ(|ξ| > ε/2) + χ(|η| > ε/2), (6.8)

where χ(A) is the indicator of event A, the left side of (6.7) can be estimatedby R1 + R2 + R3, with

R1 = E[

Z(n)(k−1)∑i=1

(X(n)ki )2χ(|X(n)

ki | > θ/2)|=(n)(k − 1)],

R2 = E[

Z(n)(k−1)∑i=1

(X(n)ki )2χ(|τ (n)

ki | > θ/2)|=(n)(k − 1)],

R3 = E[Σ′X(n)ki X

(n)kj χ(|Tn(k)| > θ)|=(n)(k − 1)],

where Σ′ is the same as in Lemma 4 and τ(n)ki = Tn(k) − X

(n)ki . It is obvious

that R1 = Z(n)(k − 1)E[(X(n)11 )2χ(|X(n)

11 | > θ/2)]. To estimate R2 we use

independence of X(n)ki and τ

(n)ki , Chebishev inequality and relation (6.5). As

the result we obtain

R2 ≤ 4b(n)

θ2

Z(n)(k−1)∑i=1

E[(τ(n)ki )2|=(n)(k−1)] =

4b2(n)

θ2Z(n)(k−1)(Z(n)(k−1)−1).

Using Cauchy-Schwarz and Chebishev inequalities and relations (6.5) and(6.6), we see that R3 is dominated by

1

θ(E[(Σ′X(n)

ki X(n)kj )2|=(n)(k − 1)]E[(Tn(k))2|=(n)(k − 1)])1/2 =

23

=

√2b3(n)Z(k − 1)

θ

√Z(k − 1)− 1.

Lemma is proved.

Proof of Theorem 3.1. In the proof we use the following version of themartingale central limit theorem from [12] (see [12], Theorem VIII, 3.33).

Theorem A. Let Unk , k ≥ 1 for each n ≥ 1 be a sequence of martingale

differences with respect to some filtration =(n)(k), k ≥ 1, such that theconditional Lindeberg condition

[nt]∑

k=1

E[(Unk )2χ(|Un

k | > ε)|=(n)(k − 1)]P→ 0 (6.9)

holds as n →∞ for all ε > 0 and t ∈ R+. Then

[nt]∑

k=1

Unk

D→ U(t) (6.10)

as n → ∞ weakly, where U(t) is a continuous Gaussian martingale withmean zero and covariance function C(t), t ∈ R+, if and only if

[nt]∑

k=1

E[(Unk )2|=(n)(k − 1)]

P→ C(t) (6.11)

as n →∞ for each t ∈ R+.

We show that the conditions of Theorem A are fulfilled by the sum in thedefinition of Yn(t). To show that (6.11) is satisfied, we use simple equality

E[(M (n)(k))2|=(n)(k − 1)] = b(n)Z(n)(k − 1) + β(n, k), (6.12)

and obtain

[nt]∑

k=1

E[(Unk )2|=(n)(k − 1)] =

b(n)

B2n(n)

[nt]∑

k=1

Z(n)(k − 1)+

+1

B2n(n)

[nt]∑

k=1

β(n, k) = Γ1(n, t) + Γ2(n, t). (6.13)

24

Note that when condition C4 is fulfilled, B2n(n) ∼ ∆2

n(n) as n →∞ . There-fore, using part (b) of Lemma 2 and Part (c) of Lemma 3, we obtain thatthe mean of Γ1(n, t) tends as n →∞ for each t ∈ R+ to ϕ(t).

Now we show that the variance of Γ1(n, t) tends to zero as n →∞. Since

V ar(Γ1(n, t)) =b2(n)

B4n(n)

V ar(

[nt]∑

k=1

Z(n)(k − 1)),

we can use equality

V ar(

[nt]∑

k=1

Z(n)(k − 1)) = R1(n, t) + 2R2(n, t), (6.14)

where

R1(n, t) =

[nt]∑

k=1

B2n(k − 1), R2(n, t) =

[nt]−2∑i=1

[nt]−1∑j=i+1

Cov(Z(n)(i), Z(n)(j)).

Let c0 6= 0. If we use Lemma 3, we obtain that as n →∞ for each t ∈ R+

R1(n, t) ∼ n3α(n)b(n)

c0

∫ t

0

να(u)du + n2β(n)

∫ t

0

∇β(u)du. (6.15)

Consider R2(n, t). Using equality

Cov(Z(n)(i), Z(n)(i + m)) = am(n)V arZ(n)(i), (6.16)

we rewrite it as

R2(n, t) =

[nt]−2∑i=1

[nt]−1∑j=i+1

aj−i(n)B2n(i) =

[nt]−2∑i=1

a−i(n)B2n(i)

[nt]−1∑j=i+1

aj(n).

We obtain due to Lemma 1 that

limn→∞

1

n

[nt]∑j=1

aj(n) =

∫ t

0

euc0du. (6.17)

The relation (6.17) shows that for any ε > 0 and sufficiently large n

R2(n, t) ≤ n

(∫ t

0

euc0du + ε

) [nt]∑i=1

a−i(n)B2n(i). (6.18)

25

Applying Lemma 3, we derive that the sum on the right side of (6.18) asn →∞ is equivalent to

n3α(n)b(n)

c0

∫ t

0

e−uc0να(u)du + n2β(n)

∫ t

0

∇β(u)du. (6.19)

Using relation (6.15) and part (b) of Lemma 2, we obtain that as n →∞b2(n)R1(n, t)

B4n(n)

∼ K1n3α(n)b3(n)

n4α2(n)b2(n)+ K2

n2β(n)b2(n)

n4α2(n)b2(n), (6.20)

where Ki, i = 1, 2, .. are various positive constants, depending on α, β, c0 andt. Similarly, using estimate (6.18) and part (b) of Lemma 2 again, we obtainthat for sufficiently large n the ratio b2(n)R2(n, t)/B4

n(n) is dominated by

K3n4α(n)b3(n)

n4α2(n)b2(n)+ K4

n3β(n)b2(n)

n4α2(n)b2(n). (6.21)

Thus, we deduce from (6.14) that when conditions C3 and C4 are satisfied,the variance of Γ1(n, t) tends to zero as n →∞ for each t ∈ R+.

It is not difficult to see that estimates (6.20) and (6.21) remain true whenc0 = 0, with only changes in constants Ki, i = 1, 2, 3, 4. Therefore, we againobtain that the variance of the first term in (6.13) tends to zero. Hence weconclude that

Γ1(n, t)P→ ϕ(t) (6.22)

as n →∞ for each t ∈ R+.The second term in (6.13) we rewrite as

Γ2(n, t) =1

B2n(n)

[nt]∑

k=1

β(k) +1

B2n(n)

[nt]∑

k=1

(β(n, k)− β(k)). (6.23)

Since nβ(n) = o(∆2n(n)) due to condition C4, using Lemma 1 we obtain that

the first term in (6.23) tends to zero. The second term is dominated by

[nt]

∆2n(n)

max1≤k≤nt

|β(n, k)− β(k)|,

and due to conditions C1 and C4 tends to zero as n → ∞. Consequently,Γ2(n, t) → 0 as n → ∞ for each t ∈ R+. Thus, applying Slutsky’s theorem,we conclude that

[nt]∑

k=1

E[(Unk )2|=(n)(k − 1)]

P→ ϕ(t) (6.24)

26

as n →∞ for each t ∈ R+, i.e. condition (6.11) is satisfied.Now we show that condition (6.9) is also satisfied. It will be satisfied if

for any ε > 0 as n →∞

1

B2n(n)

[nt]∑

k=1

E[(M (n)(k))2χ(|M (n)(k)| > εBn(n))|=(n)(k − 1)]P→ 0. (6.25)

We obtain from the definition of the process that

M (n)(k) = Tn(k) + ξ(n)k − α(n, k), (6.26)

where Tn(k) is the sum defined in Lemma 4. If we denote the expression in(6.25) by I(n, t), using (6.26) and independence of the reproduction and theimmigration processes, it can be represented as

I(n, t) = I1(n, t) + I2(n, t), (6.27)

where

I1(n, t) =1

B2n(n)

[nt]∑

k=1

E[(Tn(k))2χ(|M (n)(k)| > εBn(n))|=(n)(k − 1)],

I2(n, t) =1

B2n(n)

[nt]∑

k=1

E[(ξ(n)k )2χ(|M (n)(k)| > εBn(n))|=(n)(k − 1)],

and ξ(n)k = ξ

(n)k − α(n, k). First we estimate I1(n, t). If we use (6.26) and

inequality (6.8) for the indicator function, we obtain that it will be dominatedby I11(n, t) + I12(n, t), where

I11(n, t) =1

B2n(n)

[nt]∑

k=1

E[(Tn(k))2χ(|ξ(n)k | > εBn(n)

2)|=(n)(k − 1)],

I12(n, t) =1

B2n(n)

[nt]∑

k=1

E[(Tn(k))2χ(|Tn(k)| > εBn(n)

2)|=(n)(k − 1)].

To estimate I11(n, t), we use independence of Tn(k) and ξ(n)k , part (a) of

Lemma 4 and the Chebishev inequality, and obtain

I11(n, t) ≤ 4b(n)

ε2B4n(n)

[nt]∑

k=1

β(n, k)Z(n)(k − 1).

27

Therefore, for any δ > 0

PI11(n, t) > δ ≤ 4b(n)

δε2B4n(n)

[nt]∑

k=1

β(n, k)An(k − 1). (6.28)

The sum on the right side of (6.28) is dominated by

max1≤k≤nt

|β(n, k)− β(k)|[nt]∑

k=1

An(k − 1) +

[nt]∑

k=1

β(k)An(k − 1).

Consequently, using parts (a), (c) and (d) of Lemma 3, we derive that the

right side of (6.28) tends to zero as n →∞, which implies that I11(n, t)P→ 0

as n →∞ for each t ∈ R+.We now estimate I12(n, t). Applying Lemma 5, we obtain that

I12(n, t) ≤ S1(n, t) + S2(n, t) + S3(n, t), (6.29)

where

S1(n, t) =1

B2n(n)

[nt]∑

k=1

δ(1)n (ε/4)Z(n)(k − 1),

S2(n, t) =16b2(n)

ε2B4n(n)

[nt]∑

k=1

(Z(n)(k − 1))2,

S3(n, t) =

√8b3(n)

εB3n(n)

[nt]∑

k=1

(Z(n)(k − 1))3/2.

Obviously, we have

S1(n, t) = δ(1)n (

ε

4)

1

B2n(n)

[nt]∑

k=1

Z(n)(k − 1). (6.30)

The sum on the right side of (6.30) multiplied by b(n)/B2n(n) is equal to

Γ1(n, t), which is the first term in (6.13), and as n → ∞ converges to ϕ(t)

in probability. Taking into account conditions δ(1)n (ε) → 0 as n → ∞ and

lim infn→∞ b(n) > 0 we conclude that

S1(n, t)P→ 0 (6.31)

28

as n →∞ for each t ∈ R+. We now consider S2(n, t). For any δ > 0 we have

PS2(n, t) ≥ δ ≤ 16b2(n)

ε2δB4n(n)

R1(n, t) +16b2(n)

ε2δB4n(n)

[nt]∑

k=1

(An(k − 1))2, (6.32)

where R1(n, t) is defined in (6.14). Using (6.15) we immediately obtain thatthe first term on the right side of (4.32) tends to zero as n →∞. Using Part(a) of Lemma 2, we have

limn→∞

1

n3α2(n)

[nt]∑

k=1

(An(k − 1))2 =

∫ t

0

(µα(u))2du. (6.33)

Therefore, due to Part (b) of Lemma 2 and the fact that B2n(n) ∼ ∆2

n(n) asn → ∞, one can see that the second term on the right side of (6.32) alsotends to zero as n →∞. Hence

S2(n, t)P→ 0 (6.34)

as n → ∞ for each t ∈ R+. To estimate S3(n, t), we use the followinginequality

∑nk=1 akbk ≤ (

∑nk=1 a2

k

∑nk=1 b2

k)1/2 with

ak = Z(n)(k − 1)/B2n(n), bk = (Z(n)(k − 1))1/2/Bn(n),

and obtain that S3(n, t) ≤ (S2(n, t)Γ1(n, t)/2)1/2. Consequently, we havefrom (6.22) and (6.34) that

S3(n, t)P→ 0. (6.35)

The relations (6.28), (6.29), (6.31), (6.34) and (6.35) imply that

I1(n, t)P→ 0 (6.36)

as n →∞ for each t ∈ R+. For the second term in (6.27) we have I2(n, t) ≤Γ2(n, t), where Γ2(n, t) is defined in (6.13). Therefore,

limn→∞

I2(n, t) = 0. (6.37)

Thus, it follows from (6.27), (6.36) and (6.37) that condition (6.9) is satisfied.The assertion of Theorem 3.1 now follows from Theorem A.

29

Proof of Theorem 3.2. We again use Theorem A. Note that when C5is satisfied, B2

n(n) ∼ σ2n(n) as n → ∞. To prove that condition (6.11) is

satisfied, we again use the representation (6.13). Using Part(a) of Lemma2 and Part(c) of Lemma 3, we obtain that E[Γ1(n, t)] → 0 as n → ∞, dueto condition C5. To prove that the variance of Γ1(n, t) tends to zero, weconsider ratios in (6.20) and (6.21), with n2β2(n) in denominators instead ofn4α2(n)b2(n). We realize that the ratios tend to zero as n → ∞ when theconditions nα(n)b(n) = o(β(n)) and b(n) = o(α(n)) as n →∞ are satisfied.Hence we have

Γ1(n, t)P→ 0 (6.38)

as n → ∞ for each t ∈ R+. To estimate Γ2(n, t), we consider (6.23). UsingPart(a) of Lemma 2 and Lemma 1, we immediately obtain that the first termin (6.23) tends to

ψ(t) =1

∇β(1)

∫ t

0

uβ. (6.39)

The second term in (6.23) tends to zero, due to Part(a) of Lemma 2, Lemma1 and the second relation in the condition C1. Thus, we obtain that (6.24)holds with ψ(t) instead of ϕ(t).

Now we show that (6.25) holds under the conditions of Theorem 3.2,which implies the fulfillment of the condition (6.9). We use (6.27). Usingparts (a), (c) and (d) of the Lemma 3 as in the proof of Theorem 3.1, we

realize that the right side of (6.28) tends to zero, i.e. I11(n, t)P→ 0 as n →∞

for each t ∈ R+.Now we estimate I12(n, t) using inequality (6.29). Since B2

n(n) ∼ ∆2n(n)

under the conditions of Theorem 3.1, the relation (6.22) remains true, if wereplace B2

n(n) by ∆2n(n) in the definition of Γ1(n, t) in the proof of Theorem

3.1. Therefore, taking into account that δ(1)n (ε) ≤ b(n), we rewrite (6.30) as

S1(n, t) ≤ ∆2n(n)

B2n(n)

b(n)

∆2n(n)

[nt]∑

k=1

Z(n)(k − 1), (6.40)

which implies, due to (6.22) and the fact that ∆2n(n) = o(B2

n(n)) as n →∞,

that (6.31) holds without the condition δ(1)n (ε) → 0.

To estimate S2(n, t), we consider the inequality (6.32). Using (6.20) again,we obtain that the first term on the right side of (6.32) tends to zero. Thesecond term tends to zero due to (6.33) and the fact that n3α2(n)b2(n) =

30

o(σ2n(n)). It is easy to see that the estimate of S3(n, t) in (6.28) remains true

under the conditions of Theorem 3.2 and tends to zero as n → ∞ for eacht ∈ R+.

Now we consider the term I2(n, t) in (6.27). Using inequality (6.8) weobtain

I2(n, t) ≤ I21(n, t) + I22(n, t), (6.41)

where

I21(n, t) =1

B2n(n)

[nt]∑

k=1

E[(ξ(n)k )2χ(|Tn(k)| > εBn(n)

2)|=(n)(k − 1)],

I22(n, t) =1

B2n(n)

[nt]∑

k=1

E[(ξ(n)k )2χ(|ξ(n)

k | > εBn(n)

2)].

Taking into account independence of ξ(n)k and Tn(k) and using the Chebishev

inequality and Lemma 4, we have

I21(n, t) ≤ 4b(n)

ε2B4n(n)

[nt]∑

k=1

β(n, k)Z(n)(k − 1). (6.42)

Realizing that the right side of (6.42) and the estimate of I11(n, t) in theproof of Theorem 3.1 are the same, we can use inequality (6.28). Therefore,taking into account that n2α(n)β(n)b(n) = o(σ4

n(n)) as n →∞, we conclude

that I21(n, t)P→ 0 for each t ∈ R+.

Now we consider I22(n, t). It is obvious that

I22(n, t) ≤ 1

σ2n(n)

[nt]∑

k=1

E[(ξ(n)k )2χ(|ξ(n)

k | > εσn(n)

2)].

If we take into account that σ2n(n) ∼ Kσ2

0(n) as n → ∞, where K is a pos-itive constant depending on β and t, we see that I22(n, t) → 0 as n → ∞when the condition δ

(2)n (ε) → 0 as n →∞ is satisfied. Hence I2(n, t)

P→ 0 asn → ∞ for each t ∈ R+ and we conclude that (6.25) holds. Theorem 3.2 isthus proved.

Proof of Theorem 3.3. It is easy to see that when condition C6 is satisfied,

B2n(n) ∼ K(c0, c)n

2α(n)b(n) (6.43)

31

as n → ∞, where K(c0, c) = να(1)/c0 + c∇β(1). To prove that condition(6.11) of Theorem A is satisfied, we consider the representation (6.13). Usingthe Part(c) of Lemma 3, we have

limn→∞

E[Γ1(n, t)] =1

K(c0, c)

∫ t

0

µα(u)du. (6.44)

Looking the proof of Theorem 3.1, it is not difficult to realize that the varianceof Γ1(n, t) tends to zero as n → ∞ when the condition C6 instead of C4 issatisfied.

To estimate Γ2(n, t) we use (6.23). Using Lemma 1 and (6.44), we obtainthat the first term in (6.23) tends to

c

K(c0, c)

∫ t

0

uβdu.

The second term in (6.23) tends to zero due to the Lemma 1 and the conditionC1. Thus, condition (6.11) is satisfied with C(t) = ω(t), where

ω(t) =1

K(c0, c)

∫ t

0

(µα(u) + cuβ)du.

We now show that condition (6.9) of Theorem A is also satisfied. For this weshow that (6.25) remains true under the conditions of Theorem 3.3. Considerthe equality (6.27). The first term in (6.27) is dominated by I11(n, t)+I12(n, t)again, where I1i(n, t), i = 1, 2 are defined in the proof of Theorem 3.1. Bythe arguments, similar to those in the proof of Theorem 3.1, we obtain that

I11(n, t)P→ 0 as n → ∞ for each t ∈ R+. We now consider (6.29). Using

the estimate (6.30), we see that (6.31) remains true when δ(1)(ε) → 0 asn →∞ for each ε > 0 and lim infn→∞ b(n) > 0. Taking into account (6.44),we realize that (6.34)-(6.36) also hold under the conditions of Theorem 3.3.

To estimate the second term in (6.27), we use (6.41) in the proof ofTheorem 3.2, and see that it tends to zero in probability as n → ∞ when

δ(2)(ε) → 0 as n →∞ for each ε > 0, and consequently Yn(t)D→ W (ω(t)) as

n →∞ weakly in Skorokhod space D(R+,R).

32

Acknowledgments

This paper is based on a part of results obtained under research project NoIN080396 funded by KFUPM, Dhahran, Saudi Arabia. My sincere thanksto King Fahd University of Petroleum and Minarals for all the supports andfacilities I had.

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