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Functional Gaussian Approximation for Dependent Structures$

Florence Merlevède, Magda Peligrad, and Sergey Utev

Print publication date: 2019

Print ISBN-13: 9780198826941

Published to Oxford Scholarship Online: April 2019

DOI: 10.1093/oso/9780198826941.001.0001

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Linear Processes

Linear Processes

Chapter:
(p.345) 12 Linear Processes
Source:
Functional Gaussian Approximation for Dependent Structures
Author(s):

Florence Merlevède

Magda Peligrad

Sergey Utev

Publisher:
Oxford University Press
DOI:10.1093/oso/9780198826941.003.0012

Abstract and Keywords

Here we apply different methods to establish the Gaussian approximation to linear statistics of a stationary sequence, including stationary linear processes, near-stationary processes, and discrete Fourier transforms of a strictly stationary process. More precisely, we analyze the asymptotic behavior of the partial sums associated with a short-memory linear process and prove, in particular, that if a weak limit theorem holds for the partial sums of the innovations then a related result holds for the partial sums of the linear process itself. We then move to linear processes with long memory and obtain the CLT under various dependence structures for the innovations by analyzing the asymptotic behavior of linear statistics. We also deal with the invariance principle for causal linear processes or for linear statistics with weakly associated innovations. The last section deals with discrete Fourier transforms, proving, via martingale approximation, central limit behavior at almost all frequencies under almost no condition except a regularity assumption.

Keywords:   linear statistics, linear processes, short memory, long memory, discrete Fourier transform

In this chapter, we apply different methods to establish the Gaussian approximation to linear statistics of a stationary sequence {ξ‎i}, called the innovation sequence. Define

Xk=iaikξiand associated sumsSn=k=1nXk=iξik=1naik.

  1. (i) We first treat stationary linear processes when aij = aij.

  2. (ii) We then move to near stationary processes when the sums bin=k=1naik do not vary too much with respect to i (more precisely when they satisfy the near linearity property (12.9)).

  3. (iii) Finally, we treat special sums; namely, discrete Fourier transforms that are of the form

    Sn(θ)=j=1nXjeijθ.

12.1 Linear Processes with Short Memory

In this section we shall discuss when limiting properties of stationary processes are preserved under infinite linear transforms.

Let (ξi)iZ be a strictly stationary sequence with E(|ξ0|)< and E(ξ0)=0 and let I be its invariant σ‎–field. Define

Xk=j=ajξkjand assumei=|ai|<.
(12.1)

(p.346) The following notation is useful in various parts of the section:

Sk(ξ)=j=1kξj.

We begin with a representation of Sn in terms of partial sums and illustrate its applications to the CLT and FCLT.

We first notice that by interchanging finite and infinite sum

Sn=Sn(X)=k=1nXk=k=1nj=ajξkj=jajk=1nξkj=:jajSn,j(ξ),

where

Sn,j(ξ)=ξ1j++ξnjandSn,0(ξ)=ξ1++ξn=Sn(ξ).

Since for each finite j, Sn,j(ξ) and Sn(ξ) are identically distributed, it suggests the following intuitive approximation

Sn(X)iaiSn(ξ)=ASn(ξ),whereA=iai.

To make the argument rigorous we shall use the following simple argument:

Lemma 12.1 Let iZ|ai|< . Assume that for a sequence of positive constants bn the following conditions hold:

an,i:=E|Sn,i(ξ)Sn(ξ)|bn0for eachi,andC:=supnE|Sn(ξ)|bn<.

Then,

E|Sn(X)ASn(ξ)|bn0.

Remark 12.2 This lemma shows that if we have a weak limit theorem for the innovations, so if Sn(ξ)/bndL , then a related result holds for partial sums of linear processes with the same normalization, i.e. Sn(X)/bndAL .

Proof Notice that

Sn(X)ASn(ξ)=iZai(Sn,i(ξ)Sn(ξ))

(p.347) and thus by the triangular inequality

E|Sn(X)ASn(ξ)|iZ|ai|E|Sn,i(ξ)Sn(ξ)||i|>M|ai|supnsupiZE|Sn,i(ξ)Sn(ξ)|+iZ|ai|max|i|ME|Sn,i(ξ)Sn(ξ)|.

It remains to notice that by stationarity

E|Sn(X)ASn(ξ)|/bn2C|i|>M|ai|+iZ|ai|max|i|Man,i.

The result follows by letting n followed by M.

By using the same approach, our next proposition allows to compare the maximum of partial sums of the innovations to the maximum of partial sums of the linear process with short memory (i.e. i|ai|<) .

Proposition 12.3 Let iZ|ai|< . Assume the representation (12.1) is satisfied and in addition, there is a constant C > 0 and a sequence of positive reals bn such that for all n,

Emax1jn|Sj(ξ)|Cbn
(12.2)

and

bn1max1jn|ξj|P0asn.
(12.3)

Then,

1bn(max1jn|Sj(X)ASj(ξ)|)P0asn.
(12.4)

If the innovations are assumed in Lp , p ≥ 1, (12.2) is replaced by Emax1jn|Sj(ξ)|pCbnp and the convergence in (12.3) holds in Lp , then

1bnpE(max1jn|Sj(X)ASj(ξ)|p)0asn.

Proof The proof is similar. Note that, for any positive integer M less than n,

max1mn|Sm(X)ASm(ξ)|iZ|ai|max1mn|Sm,i(ξ)Sm(ξ)||i|>M|ai|(max1mn|Sm,i(ξ)Sm(ξ)|)+iZ|ai|max|i|Mmax1mn|Sm,i(ξ)Sm(ξ)||i|>M|ai|(max1mn|Sm,i(ξ)Sm(ξ)|)+2MiZ|ai|(maxMin+M|ξi|):=I+II.

(p.348) For M fixed the second term II divided by bn converges to 0 in probability as n by condition (12.3).

Next, by Markov inequality and stationarity, for any ε‎ > 0,

P(I>εbn)2(εbn)1|i|>M|ai|E(max1mn|Sm(ξ)|)Cε1|i|>M|ai|,

which converges to 0 when M. So the first part of the proposition follows by letting first n followed by M. The proof of the second part of the proposition is done by using similar arguments.

Theorem 12.4 Let iZ|ai|< . Assume that representation (12.1) and condition (12.2) are satisfied. Moreover, assume that the innovations satisfy the invariance principle {bn1S[nt](ξ),t[0,1]}dηW in D([0, 1]) as n , where η‎ is I -measurable and W is a standard Brownian motion on [0, 1] independent on I . Then the linear process also satisfies the invariance principle, i.e. {bn1S[nt](X),t[0,1]}dηAW in D([0, 1]) as n .

Proof Notice that the convergence in probability in (12.3) follows from the invariance principle {bn1S[nt](ξ),t[0,1]}dηW in D([0, 1]) as n , since the modulus of continuity is convergent to 0 in probability. All the conditions in Proposition 12.3 are then satisfied which imply the conclusion of the theorem.

From Theorem 12.4 we easily derive the following useful consequence.

Corollary 12.5 ( Lp –invariance principle.) Let iZ|ai|< . Assume the representation (12.1) holds and p ≥ 1. Then,

limnE(sup0t1|bn1S[nt](ξ)ηW(t)|p)=0limnE(sup0t1|bn1S[nt]ηAW(t)|p)=0.

Discussion

Theorem 12.4 and its Corollary 12.5 work for many dependent structures such as surveyed in Peligrad (1986), Doukhan (1994), Bradley (2007) and Merlevède, Peligrad and Utev (2006). Various invariance principles can be extended from the original sequence to the linear process with short memory. Here we mention some traditional and also some recently developed dependence conditions for innovations whose partial sums satisfy both a maximal inequality and the invariance principle and therefore Theorem 12.4 and its Corollary 12.5 apply.

(p.349) For instance, let us assume that (ξi)iZ is a stationary ergodic sequence with E(ξ02)< and E(ξ0)=0 and let Fk=σ(ξi,ik) . For all the structures below the family (max1kn(Sk(ξ))2/n)n1 is uniformly integrable and the conclusion of Corollary 12.5 holds with bn=n and with p = 2. Moreover, since (ξi)iZ is assumed to be ergodic then there is a non-negative constant σ‎ such that η‎ = σ‎.

  1. (i) Hannan (1979) (see also its extension to Hilbert space in Dedecker and Merlevède (2003)):

    n=1P0(ξn)2<andE(ξ0|F)=0a.s.

    where Pk(X)=E(X|Fk)E(X|Fk1) is the projection operator (see Theorem 4.17).

  2. (ii) Newman and Wright (1981): (ξi)iZ is an associated sequence (i.e. such that any two coordinatewise non-decreasing functions of any finite subcollection of the ξ‎i’s (of finite variance) are non-negatively correlated) which satisfies in addition

    k=1cov(ξk,ξ0)<.

  3. (iii) Doukhan, Massart and Rio (1994):

    k=0(k+1)0α(k)Q2(u)du<,

    where Q denotes the càdlàg inverse of the function tP(|ξ0|>t) and (α‎(k))k≥0 is the sequence of strong mixing coefficients associated with (ξi)iZ (see Theorem 6.39).

  4. (iv) Dedecker and Rio (2000):

    E(ξ0Sn(ξ)|F0)converges inL1.

    (See Theorem 4.18).

  5. (v) Peligrad and Utev (2005), by developing Maxwell and Woodroofe (2000),

    n=1E(Sn(ξ)|F0)2n3/2<.

    (See Theorem 4.16).

  6. (vi) Peligrad, Utev and Wu (2007), which guarantees the Lp -invariance principle, p ≥ 2.

    (p.350)

    n=1E(Sn(ξ)|F0)pn3/2<.

Comments

  1. (a) If E(ξ02)< and bnn then condition (12.3) automatically holds.

  2. (b) The set of indexes Z can be replaced by Zd where d is a positive integer allowing for the treatment of random fields.

  3. (c) A natural extension is to consider innovations with values in functional spaces that also facilitate the study of estimation and forecasting problems for several classes of continuous time processes (see Bosq (2000)). The linear processes are still defined by the formula (12.1) with the difference that now the innovations (ξk)kZ take values in a separable Hilbert space H and the sequence of constants is replaced by the sequence of bounded linear operators {ak}kZ from H to H. Merlevède, Peligrad and Utev (1997) treated the problem of the central limit theorem for this case under the summability condition

    j=ajL(H)<,

    where ∥ajL(H) denotes the usual operators norm. It was discovered that, if this condition is not satisfied, then the central limit theorem fails even for the case of independent innovations. The approach developed in that paper shows that the central limit theorem results stated can be strengthened to the invariance principle (some results in this direction for strongly mixing sequences are established in Merlevède (2003)).

  4. (d) In all the examples (i)–(iv) the variance of partial sums is linear in n.

12.2 Functional CLT using Coboundary Decomposition

Here we shall establish a functional CLT by using the coboundary decomposition.

Lemma 12.6 Assume that the sequence of innovations (ξj)jZ is strictly stationary, centered with finite second moment and has a bounded spectral density. Assume in addition that the sequence of constants (ak)k≥0 satisfies the following two conditions

A=k=0akis convergent andj0k=j+1ak2<.

For any kZ , define the causal linear process

Xk=j0ajξkj.

(p.351) Then there is a stationary sequence (ν‎k)k of square integrable real-valued random variables such that, for any Z , the following coboundary decomposition holds:

X=Aξ+ν1ν.
(12.5)

Proof Let us introduce the stationary sequence

ν=j0k=j+1akξj,Z.

This sequence is well defined in L2 under the conditions of this theorem. With this definition of ν‎, it is immediate that (12.5) is satisfied.

As an immediate consequence we obtain:

Theorem 12.7 Assume that the sequence of innovations and of constants satisfy the conditions of Lemma 12.6. Moreover assume that the innovations satisfy the invariance principle {bn1S[nt](ξ),t[0,1]}dηW in D([0, 1]) as n , where η‎ is I -measurable and W is a standard Brownian motion on [0, 1] independent on I . Suppose also that

max1n|ν|bn0.

Then the linear process also satisfies the invariance principle, i.e. {bn1S[nt](X),t[0,1]}dηAW in D([0, 1]) as n .

Proof The functional CLT follows from the corresponding one for the innovations by noticing that, for any t ∈ [0, 1],

S[nt](X)=AS[nt](ξ)+ν0ν[nt].

12.3 Toward Linear Processes with Long Memory

We consider in this section linear processes Xj=kZajkξk . In many situations the process {Xj,jZ} is well defined under weaker conditions that (12.1). For instance, when the innovations form an i.i.d. sequence of centered random variables which are also square integrable the necessary and sufficient condition for the existence of X0 in the almost sure sense is

iZai2<
(12.6)

(implied by |ai|< ). So, whenever X0 is well defined we can represent the partial sums as a linear statistic

(p.352)

Sn=kξki=1naik:=kZξkbk,n,

where bk, n := a1−k + … + ank.

It should be noted that in this case the variance of partial sums might not be asymptotically linear in n. As a matter of fact, if the innovations are i.i.d. centered with finite second moment the variance of Sn can range from a constant to close of n2. When the variance is not asymptotically linear in n we shall refer to that case as long memory.

We shall obtain first the central limit theorem for the linear statistic for various dependence structures that will allow us to obtain the CLT for the classes of linear processes satisfying condition (12.6). The invariance principle is much more delicate in the long memory case. The discussion of the invariance principle will follow the section on the central limit theorem.

12.3.1 CLT for Linear Statistics with Dependent Innovations via Martingale Approximation

Let (ξi)iZ be a strictly stationary sequence of centered real-valued random variables with finite second moment. The aim of this section is to study the asymptotic behavior of linear statistics of the type

Sn=kZbk,nξk,
(12.7)

where {bk,n,kZ} is a triangular array of numerical constants satisfying kZbk,n2< for any n. Note that the linear statistic Sn is properly defined if and only if the stationary sequence of innovations (ξi)iZ has a bounded spectral density.

The main result of this section is the following:

Theorem 12.8 Let (ξi)iZ be a strictly stationary and ergodic sequence of real-valued random variables with finite second moment, centered at expectations, adapted to a stationary filtration (Fi)iZ and satisfying

Γj=k=0|E[ξkE(ξ0|j)]|<for all jand1pj=1pΓj0asp.
(12.8)

Let {bk,n,kZ} be a triangular array of numerical constants such that

bn2=kZbk,n2<andlimn1bn2kZ(bk,nbk1,n)2=0.
(12.9)

(p.353) Let Sn be defined by (12.7). Then, (ξi)iZ has a continuous spectral density f,

limnVar(Sn)bn2=2πf(0)andSnbnd2πf(0)Nasn,
(12.10)

where N is a standard normal variable.

Comment 12.9 This result is due to Peligrad and Utev (2006). We note that if we do not assume the sequence (ξi)iZ to be ergodic then it can be proven that there is a non-negative random variable η‎2 measurable with respect to I (the invariant σ‎-field) such that n1E((k=1nξk)2|F0)η2 in L1 as n and E(η2)=2πf(0) . In this situation, the second part of (12.10) has to be modified as follows: bn1SndηN as n , where N is a standard normal variable independent of η‎. For a complete proof of this comment, we refer the reader to the proof of Theorem 1 in Peligrad and Utev (2006).

As a corollary of Theorem 12.8, we can derive the following central limit theorem for the partial sums associated with a linear process {Xk,kZ} defined by

Xk=jZakjξj,kZ,
(12.11)

when jZaj2< and (ξi)iZ satisfies the conditions of Theorem 12.8.

Corollary 12.10 Let (ξi)iZ be as in Theorem 12.8 and (ak)kZ be a sequence of real numbers such that jZaj2< . Let

bn2=kZbk,n2:=kZ(a1k++ank)2.
(12.12)

and assume that bn2 as n. Let {Xk}kZ be defined by (12.11). Then

bn1k=1nXkd2πf(0)Nasn,

where N is a standard normal variable and f is the spectral density of (ξi)iZ .

Indeed, recall the representation

Sn=kZξki=1naik=:kZξkbk,n,

where bk, n := a1−k + … + ank. Observe that, by the Cauchy inequality,

bk,n2=a1k++ank2n(a1k2++ank2)

(p.354) and so the series

kZbk,n2niZ(a1i2++ani2)=n2iZai2

which is certainly finite if iZai2< . To end the proof of the corollary, observe that

jZ(bj,nbj1,n)24kZak2.

Comment 12.11 Condition (12.8) is satisfied under various different dependent conditions. We just mention in what follows that it holds if (ξi)iZ satisfies Hannan’s condition (4.29), namely:

E(ξ0|F)=0a.s. andi0P0(ξi)2<,

where we recall that P0()=E(|F0)E(|F1) . Indeed, since E(ξ0|F)=0 a.s., we have the following representation:

ξk=i=kPi(ξk)andE(ξ0|Fj)=i=jPi(ξ0).

By stationarity ∥Pn(ξ‎0)∥2 = ∥Pn+k(ξ‎k)∥2 for any kZ . Next, Pi(ξ‎0) and Pj(ξ‎k) are uncorrelated for ij, implying that

E[ξkE(ξ0|Fj)]=i=min(k,j)EPi(ξk)Pi(ξ0).

As a consequence

|E[ξkE(ξ0|Fj)]|i=jPi(ξk)2Pi(ξ0)2=i=jPik(ξ0)2Pi(ξ0)2.

Therefore,

Γj =k=0|E[ξkE(ξ0|j)]|i=jPi(ξ0)2k0Pk(ξ0)2=ijP0(ξi)2k0P0(ξk)2.

Whence, under Hannan’s condition, we derive that limjΓj=0 , proving the validity of condition (12.8). We refer to the paper by Peligrad and Utev (2006) for other dependence conditions implying condition (12.8).

(p.355) Proof of Theorem 12.8 Since by assumption, k=0|E[ξkξ0]|=Γ0< , by Lemma 1.5, (ξi)iZ has a continuous spectral density and the first part of (12.10) holds. The proof of the second part of (12.10) will be divided in two steps. In step 1, we prove that, under the additional assumption that (ξi)iZ forms a sequence of martingale differences then the conclusion of Theorem 12.8 is satisfied. In the second step, and with the help of step 1, we use a martingale approximation to show that Theorem 12.8 holds in its full generality under condition (12.8).

Step 1. In this step, we shall assume that (ξi)iZ is a strictly stationary and ergodic sequence of martingale differences with finite second moment, and prove that, in this case SndN(0,E(ξ02)) .

We shall apply Corollary 2.32 to the triangular array of martingale differences (dn,j)jZ,n1 where, for any jZ and any n ≥ 1,

dn,j=bn1(bj,nξj).

Note first that, for any positive integer n, bn2jZE(bj,nξj)2=E(ξ02) . The conclusion of the theorem will then follow from Corollary 2.32 if one can prove that the triangular array {dn,j,jZ}n1 satisfies the conditions (2.44) and (2.45).

To verify (2.44), that is: bn1E(supkZ|bk,nξk|)0 as n , it is enough to check the Lindeberg condition: For any ε‎ > 0,

bn2iZE[bi,n2ξi21{bn1|bi,nξi|>ε}]0,asn.
(12.13)

But, by stationarity, for any ε‎ > 0,

bn2iZE[bi,n2ξi21{bn1|bi,nξi|>ε}]maxiZEξ121{bn1|bi,nξ1|>ε}Eξ121{bn1maxiZ|bi,n||ξ1|>ε}.

We then conclude that (12.13) will hold if bn1maxjZ|bj,n|0 . This latter condition holds under condition (12.9) as stated in the following lemma about sequences, whose proof will be given at the end of the proof of the theorem.

Lemma 12.12 Under (12.9), limnbn1maxkZ|bk,n|=0 .

It remains to show that the triangular array {dn,j,jZ}n1 satisfies condition (2.45). So we shall show in what follows that the following convergence holds:

Jn:=1bn2j=bj,n2ξj2PE(ξ02)asn.
(12.14)

The approach to this proof is to make blocks of variables and replace each variable in a block by the average of variables forming the block. The following lemma, with the Hilbert space type language, is convenient.

(p.356) Lemma 12.13 Let 2 be a Hilbert space of double sequences x={xj}jZ with the norm x22=j|xj|2 . Set also x1=j|xj| . Let the translation operator be denoted by Tx(j) = xj+1. Given a fixed positive integer p, by Ik denote the set of integers Ik = {(p(k − 1) + 1, …, kp} and associate the sequence {(Apx)j}jZ based on the average of the terms in a block:

(Apx)j=1piIkxi,forjIk.

In addition, define (x2)k=(xk2) . Then, for any positive integers j and p,

xTjx2jxTx2,xApx2pxTx2andx2Apx212pxTx1,x2y21xy22+2xy2x2.

In particular, assume that we have a sequence of elements x(n)2 such that ∥x(n)2 = 1 and x(n)Tx(n)20 as n . Then, for any positive integers j and p,

x(n)2Tjx(n)210,x(n)2(Apx(n))210andx(n)2Ap(x(n)2)10,

as n .

Proof The proof requires easy algebraic manipulations and is left to the reader.

With the help of this lemma, let us prove (12.14). Fix a positive integer p and make small blocks of normalized sums of consecutive random variables. Define

(x(n))i=xi,n=bi,n/bn,tk,n=(Ap(x(n)2))j=p1iIkxi,n2forjIk,

and decompose the sum in (12.14) in the following way

Jn=j=xj,n2ξj2=k=ptk,n(1ppkj=p(k1)+1ξj2)+k=pkj=p(k1)+1[xj,n2tk,n]ξj2:=Jn,1+Jn,2.

Notice first that Σkptk,n=bn2 and as a consequence, by stationarity and the L1 ergodic theorem (see Theorem 1.3), the following convergence holds uniformly in n

E|Jn,1E(ξ02)|E1pj=p(k1)+1pkξj2E(ξ02)0asp.

(p.357) On the other hand,

E|Jn,2|E[ξ02]k=j=p(k1)+1pk|xj,n2tk,n|=E[ξ02]x(n)2Ap(x(n)2)10asn,

by Lemma 12.13. This ends the proof of (12.14) and then of Step 1.

Step 2. We show now that Theorem 12.8 holds in its full generality. So, we assume from now on that (ξi)iZ is a strictly stationary and ergodic sequence of centered real-valued random variables with finite second moment that satisfies condition (12.8).

This step is based on a blocking procedure and then on an approximation of the sums of variables in blocks by martingale differences. As before, let p be a fixed positive integer and denote by Ik = {(k − 1)p + 1, …, kp}. So Ik’s are blocks of consecutive integers of size p and Z=k=Ik . Let

xj,n=bj,nbn,ck,n=1piIkxi,n.

We start with the following decomposition

Wn=j=xj,nξj=k=jIkxj,nξj=k=pck,n1pjIkξj+k=jIk[xj,nck,n]ξj:=Bn,1+Bn,2.

We shall show first that Bn, 2 is negligible for the convergence in distribution. As noticed at the beginning of the proof, (ξi)iZ has a continuous spectral density and by the second inequality in part (i) of Lemma 1.5, the variance of Bn, 2 is bounded by

E(Bn,22)E[ξ02]+2k=1|E(ξ0ξk)|k=jIk(xj,nck,n)2.

On the other hand, notice that, by Lemma 12.13,

k=jIk(xj,nck,n)2=x(n)Apx(n)220asn,

which proves that

E(Bn,22)0asn.

(p.358) To analyze Bn, 1 we denote the weighted sum in a block of size p by

Yk(p)=1pjIkξj,kandGk=kp.

Then, Yk(p) is Gk -measurable and define

Zk(p)=E(Yk(p)|Gk1)andVk(p)=Yk(p)Zk(p).

Obviously Vk(p) is a stationary sequence of martingale differences and Yk(p)=Zk(p)+Vk(p). It follows that Bn, 1 can be decomposed into a linear process with stationary martingale differences innovations and another one involving Zk(p).

We shall show first that the term involving Zk(p) is negligible for the convergence in distribution in the sense that

limplimsupnk=pck,n(Zk(p))22=0.
(12.15)

Let δn,k=pck,n . Observe that δn2:=kZδn,k21 . In addition,

kZδn,k2=k=pck,n2=1+k=jIk(ck,n2xj,n2),

and, by Lemma 12.13 and the construction,

k=jIk|xj,n2ck,n2|=x(n)2(Apx(n))210asn.

Hence,

kZδn,k21asn.

Moreover,

kZ|δn,kδn,k1|2p4bn2jZ|bj,nbj1,n|2.

Hence, since (12.9) is assumed, the sequence {δ‎n, k} satisfies condition (1.11). Therefore, according to Lemma 1.5, part (iii), we deduce that

limnk=pck,n(Zk(p))22=2πf(p)(0),

(p.359) where f(p)(x) denotes the spectral density of the process {Zk(p)}kZ . On the other hand, since

f(p)(0)=limn1nk=1n(Zk(p))22,

in order to establish (12.15) it is enough to show that

limpk=1|E(Z1(p)Zk(p))|=0.

First, we observe that, for any k ≥ 1,

|E(Z1(p)Zk(p))|=1pEEi=1pξiF0j=(k1)p+1kpξj.

By the triangle inequality and Condition (12.8) obviously

k=1|E(Z1(p)Zk(p))|21pi=1pn=i|E[E(ξi|0)ξn]|21pi=1pΓi0as p.

To complete the prove we have to show that the remaining linear process involving the martingale differences satisfies the desired CLT. We shall denote by

Xk(p)=j=pck,nVj(p)andSn(p)=k=1nXk(p).

Since the coefficients δn,k=pck,n satisfy (1.11), by step 1, it follows that for any p fixed,

Sn(p)E([V1(p)]2)Nasn.

In order to complete the proof, by Theorem 1.10 we have only to establish that

E([V1(p)]2)2πf(0)asp.

With this aim, we notice that, by stationarity,

E([V1(p)]2)=p1Ei=1pξiE(ξi|F0)2=E(ξ02)+2p1i=1p1(pi)E(ξ0ξi)p1i,j=1pEξiE(ξj|F0).

(p.360) Since Γ0< ,

limpE(ξ02)+2p1i=1p1(pi)E(ξ0ξi)=2πf(0).

Therefore, the proof will be complete if one can prove that

limpp1i,j=1pEξiE(ξj|F0)=0.
(12.16)

But, by stationarity,

1p|i,j=1pE(ξiE(ξj|0))|2pi=1pk0|E(ξkE(ξ0|i))|:=2pi=1pΓi,

proving (12.16) by taking into account condition (12.8). This ends the proof of step 2. To end the proof of Theorem 12.8, it remains to prove Lemma 12.12.

Proof of Lemma 12.12 To prove it, we proceed as follows. Let m be a fixed positive integer. Note first that if i0 is such that |bi0,n|=maxkZ|bk,n| then, since i0 = j0 (mod(m)) where j0 ∈{0, …, m − 1}, it follows that

maxkZ|bk,n|maxj{0,,m1}maxkZ|bkm+j,n|.

It follows that

maxkZbk,n4j=0m1kZbkm+j,n4j=0m1kZbkm+j,n22.

Therefore, the lemma will follow if we can prove that

limmlimsupn1bn4j=0m1kZbkm+j,n22=0.
(12.17)

With this aim, observe that, for any j = 0, …, m − 1,

kZbkm+j,n21mkZbk,n2=kZbkm+j,n21m=0m1kZbkm+,n21m=0m1kZbkm+j,n2bkm+,n21m=0j1kZu=+1jbkm+u,n2bkm+u1,n2+1m=jm1kZu=j+1bkm+u,n2bkm+u1,n2,

(p.361) implying that

kZbkm+j,n21mkZbk,n2u=1m1kZbkm+u,n2bkm+u1,n2iZbi,n2bi1,n2.

Therefore,

1bn4j=0m1kZbkm+j,n222mbn4iZbi,n2bi1,n22+2mbn4kZbk,n228mbn2iZbi,nbi1,n2+2m,

which proves (12.17) by taking into account condition (12.9).

12.4 Invariance Principle for Linear Processes

We move now to explore the invariance principle for the partial sums associated with a causal linear process defined by

Xk=j=0ajξkj,kZ.
(12.18)

Assume that X0 exists, is in L2 and is defined as before. Let us associate with the partial sums the following process in D([0, 1])

Wn(t)=j=1[nt]Xj/σn,t[0,1],whereσn2=Var(Sn).

We would like to find the limiting distribution of this process when a CLT is available. It should be noted that if {Wn(t), t ∈ [0, 1]} converges weakly to a standard Brownian motion, then necessarily σn2=nh(n) where h(n) is a slowly varying function (i.e. a regularly varying function with exponent 1). This is so since for t ∈ [0, 1] fixed we have S[nt]/σndN(0,t) and, in addition, taking t = 1 we have Sn2/σn2 is uniformly integrable (by the convergence of moments theorem), implying σ[nt]2/σn2t .

A natural conjecture is that if we assume that the innovations (ξ‎k) are i.i.d. centered with finite second moment E(ξ02)>0 , j=0aj2< and σn2=nh(n) with h(n) a slowly varying function, then {Wn(t), t ∈ [0, 1]} converges weakly to a standard Brownian motion. This conjecture however has a negative answer.

(p.362) 12.4.1 Construction of the Counterexample

Let us first give a characterization for the variance of the partial sums of a linear process with i.i.d. innovations to be linearly varying with exponent 1 (see Definition 12.17).

Lemma 12.14 Let (ξi,iZ) be a sequence of i.i.d. centered real-valued r.v.’s with finite second moment E[ξ12]=σ2>0 . In addition, let (ai, i ≥ 0) be a sequence of real numbers such that i0ai2< . Then we consider the causal linear process defined by (12.18). Let bn=a0++an and assume that

k=0nbk2,asn,
(12.19)

and

j=0(bn+jbj)2=ok=0n1bk2.
(12.20)

Then, k=0n1bk21σn2σ2 and σn2=nh(n) where h(n) is a slowly varying function.

Remark 12.15 Notice that Wu and Woodroofe (2004) pointed out that (12.20) is a necessary and sufficient condition in order for (4.86) to hold for the sequence (Xk)kZ .

Proof Notice that

Sn=j=1nbnjξj+j=0(bn+jbj)ξj

and then

σn2=σ2k=0n1bk2+j=0(bn+jbj)2.
(12.21)

This shows that k=0n1bk21σn2σ2 . Let now Fk=σ(ξ,k) and note that

E(Sn|F0)=j=0(bn+jbj)ξj.

Then E(E(Sn|F0))2=σ2j=0(bn+jbj)2 . Notice that our conditions imply

σn2andσn1E(Sn|F020,asn.

The result follows by applying Proposition 4.30.

(p.363) The next example shows that conditions (12.19) and (12.20) are not sufficient to ensure that linear processes (Xk)kZ , as defined above, satisfy the weak invariance principle.

Proposition 12.16 There exist a sequence of i.i.d. innovations {ξ‎i}, centered with positive and finite second moments and a sequence (ai, i ≥ 0) of real numbers, satisfying i0ai2< , such that the linear process (Xk)kZ defined by (12.18) satisfies σn2 = nh(n) with h(n) slowly varying and such that the weak invariance principle does not hold.

Proof of Proposition 12.16 Our example is inspired by the construction of examples as in Herrndorf (1983), and also by the paper of Wu and Woodroofe (2004). Let us define two sequences {an, n ≥ 0} and {an,n0} as follows:

a0=0,a1=1log2andan=1log(n+1)1logn,forn2

and

a0=1,a1=1,a2=0andan=an2,forn3.

Let now (ξi,iZ) be a sequence of independent, identically distributed and symmetric random variables such that

P(ξ0>x)1x2log3/2x,forxlarge enough.
(12.22)

Define now two linear processes:

Xk=j0ajξkjandXk=j0ajξkj.

Denote σn2=E(k=1nXk)2 and σn2=E(k=1nXk)2 . Let bn=a0++an . Since

j=0(bn+jbj)2=Onlog3nandk=0n1bk2n(logn)2,

it follows that both of the linear processes {Xk,kZ} and {Xk,kZ} satisfy the conditions of Lemma 12.14.

Now observe that

Xk+2=Xk+ξk+2ξk+1,

then: k=1nXk+2=k=1nXk+ξn+2ξ2 . It follows that σn2σn2 . In addition, according to (12.22), classical computations yield that for every ε‎ > 0,

limsupnP(max1kn|ξk+2ξk+1|εn/logn)=1.
(12.23)

(p.364) As a consequence, it follows that the sequences of processes {{σn1i=1[nt]Xi,t[0,1]} and {(σn)1i=1[nt]Xi,t[0,1]} cannot satisfy the weak invariance principle at the same time. Indeed, if for instance the sequence {σn1i=1[nt]Xi,t[0,1]} satisfies the weak invariance principle, then necessarily for every ε‎ ≥ 0, P(max1in|Xi|εσn)0 , as n , and consequently, from (12.23),

limsupnP(max1in|Xi|εσn)limsupnP(max1kn|ξk+2ξk+1|εn/logn)=1.

Then, for linear processes, the weak invariance principle cannot hold without additional assumptions to the conditions of Lemma 12.14.

12.4.2 Finite-dimensional Distributions

In this subsection we shall describe the behavior of the finite-dimensional distributions of the process

Wn(t)=S[nt]/bn,0t1

where the sequence of innovations, constants and notations are as in Corollary 12.10.

Definition 12.17 We say that a positive sequence (vn2)n1 is regularly varying with exponent β‎ > 0 if, for any t ∈[0, 1],

v[nt]2vn2tβ,asn.
(12.24)

Definition 12.18 A Gaussian process is called a standard fractional Brownian motion on [0, 1] with Hurst index α‎ ∈ (0, 1) if it has the following covariance structure: For any s, t ∈ [0, 1],

Cov(WH(s),WH(t))=12(s2α+t2α|ts|2α).

Theorem 12.19 Assume the innovations and the constants satisfy the conditions of Corollary 12.10. Let β‎ ∈]0, 2] and assume that bn2 is regularly varying with exponent β‎. Then the finite-dimensional distributions of {bn1S[nt],t[0,1]} converge to the corresponding ones of 2πf(0)WH , where WH is the standard fractional Brownian motion with Hurst index H = β‎/2.

Example 1 Let us consider the linear process Xk defined by

Xk=(1B)dξk=i0aiξkiwithai=Γ(i+d)Γ(d)Γ(i),
(12.25)

where 0 < d < 1/2, B is the lag operator, and (ξi)iZ is a strictly stationary sequence satisfying the condition of Theorem 12.8. Then Theorem 12.19 applies with β‎ = 2d + 1, since akκ‎dkd−1 for some κ‎d > 0.

(p.365) Example 2 Now, if we consider the following choice of (ak)k≥0: a0 = 1 and ai = (i+1)α‎iα‎ for i ≥ 1 with α‎ ∈]0, 1/2[, then the theorem also applies. Indeed for this choice, bn2καn12α , where κ‎α‎ is a positive constant depending on α‎.

Example 3 For the selection aiiα‎(i) where is a slowly varying function at infinity and 1/2 < α‎ < 1 then, bn2καn32α2(n) (see for instance Relations (12) in Wang et al. (2001)), where κ‎α‎ is a positive constant depending on α‎.

Example 4 Finally, if aii1/2(logi)α for some α‎ > 1/2, then bn2n2(logn)12α/(2α1) (see again Relations (12) in Wang et al. (2001)). Hence (12.24) is satisfied with β‎ = 2.

Proof of Theorem 12.19 To prove the convergence of the finite-dimensional distributions, we shall apply the Cramér–Wold device. Let m be a positive integer. Let 0<t1<t2<<tm1 and set n = [nt]. For λ1,,λmR , notice that

=1mλSnbn=1bnjZ=1mλbj,nξj,
(12.26)

where bj, n = a1−j + ⋯ + anj for all jZ , and bn2=jZbj,n2 .

We shall apply Theorem 12.8 to the linear process jZBj,nξj where

Bj,n==1mλbj,n.
(12.27)

As a first step we calculate the limit over n of the following quantity

1bn2jZBj,n2=1bn2jZ=1mk=1mλλkbj,nbj,nk.

For any 1 ≤ km, by using the fact that for any two real numbers A and B we have A(A + B) = 2−1(A2 + (A+B)2B2), we get that

1bn2jZbj,nbj,nk=12bn2jZbj,n2+bj,nk2bj,nbj,nk2=12bn2jZbj,n2+bj,nk2bj,nkn2.

Now, by using condition (12.24), we derive that, for any 1 ≤ km,

jZbj,nbj,nkbn212tβ+tkβ(tkt)β.
(12.28)

(p.366) It follows from (12.28) that

limn1bn2jZBj,n2=12=1mk=1mλλktβ+tkβ|tkt|β.
(12.29)

Moreover,

1bn2jZ(Bj,nBj1,n)2=1mλ2k=1m1bn2jZ(bj,nkbj1,nk)20,asn,

since jZ(bj,nkbj1,nk)24iZai2 and bn . Therefore, the conditions of Theorem 12.8 being satisfied, using (12.29), we can deduce that

=1mλSnbnconverges in distribution toΛm,β2πf(0)N,

where

Λm,β=12=1mk=1mλλktβ+tkβ|tkt|β

ending the proof of the convergence of the finite-dimensional distributions.

12.4.3 Tightness

We shall comment about the functional CLT only for i.i.d. innovations.

Theorem 12.20 Let (ai)iZ be in 2 and let (ξi)iZ be i.i.d. centered real-valued random variables with ∥ξ‎02 = 1. Let β‎ ∈]0, 2] and assume that σn2 (the variance of Sn) is regularly varying with exponent β‎. If β‎ ∈]1, 2] then the process {σn1S[nt],t[0,1]} converges in distribution in D([0, 1]) to WH where WH is a standard fractional Brownian motion with Hurst index H = β‎/2. If β‎ ∈]0, 1] and we assume in addition that, for a real q > 2/β‎, we have ξ0q< , then the process {σ‎n−1S[nt], t ∈ [0, 1]} converges in distribution in D([0, 1]) to WH.

Proof By Theorem 12.19, it is enough to prove tightness. With this aim, we apply Corollary 1.22. By the Khintchine–Burkholder inequality applied to linear statistics of i.i.d. variables {ξi}iZ , we get that, if ξ0q< for some q ≥ 2, then

E|Sn|q=EjZbj,nξjqcE|ξ1|qjZbj,n2q/2=cE|ξ1|qσnq,

where bj, n = a1−j + … + anj. Recall that σn2=nβh(n) where h is slowly varying. Therefore by Corollary 1.22, the tightness follows by considering either q > 2/β‎ if β‎ ∈]0, 1] or q = 2 if β‎ ∈]1, 2].

(p.367) 12.5 IP for Linear Statistics with Weakly Associated Innovations

12.5.1 The Case of Asymptotically Negative Dependent Innovations

From Corollary 9.14, we can prove the following result:

Corollary 12.21 Let ξ=(ξk)kZ be a L2 -stationary sequence of real-valued centered random variables with a continuous spectral density function f on (−π‎, π‎], satisfying the asymptotic negative dependence condition (9.2) and such that {ξk2} is an uniformly integrable family. Assume in addition that ∥ξ‎02 = 1. Consider a triangular array of non-negative numerical constants {bk,n,kZ} satisfying the condition (12.9). Define the triangular array {Xk,n,kZ}n1 by

Xk,n:=bk,nξk/bn.

For 0 ≤ t ≤ 1 we set

kn(t)=minkZ:i=kE[Xi,n2]t

and Wn(0) = 0, Wn(t)=i=kn(t)Xi,n. Then {Wn(t),t[0,1]}dV in D([0, 1]) where V=2πf(0)W with W the standard Brownian motion. In particular

1bnkZbk,nξkd2πf(0)N,

where NN(0,1) .

Proof of Corollary 12.21 To prove the corollary, we shall take into account Comment 9.17 and apply Corollary 9.14 to the triangular array {Xk,n,kZ}n1 . Since the sequence (ξk)kZ is asymptotically negative dependent and the coefficients bk, n are non-negative then the triangular array {Xk,n,kZ}n1 is also asymptotically negative dependent. Next, for any ε‎ > 0,

kZE(Xk,n21{|Xn,k|>ε})supkZE(ξk21{|ξk|>δn1ε}),

where δn=bn1maxkZ|bk,n| . But by Lemma 12.12, since (12.9) is assumed, limnδn=0 . This convergence, together with the uniform integrability of {ξk2} entail that {Xk,n,kZ}n1 satisfies the Lindeberg condition.

To apply Corollary 9.14 (see also Remark 9.15), it remains to show that for all 0 ≤ a < b

Var(Wn(b)Wn(a))2πf(0)(ba).
(12.30)

(p.368) Note that

Var(Wn(b)Wn(a))=Ei=kn(a)+1kn(b)Xi,n2=bn2Ek=kn(a)+1kn(b)bk,nξk2.

Using the fact that (ξk)kZ is L2 -stationary with a continuous spectral density function f on (−π‎, π‎] and proceeding as in the proof of point (iii) of Lemma 1.5, we infer that (12.30) will hold if one can prove that, for any iZ ,

limnAn,i(ba)=0,
(12.31)

where

An,i=bn2k=kn(a)+1kn(b)bk,nbk+i,n.

To prove the convergence (12.31), note that for any iZ ,

An,i=bn2k=kn(a)+1kn(b)bk,n2+bn2k=kn(a)+1kn(b)bk,n(bk+i,nbk,n).

Hence,

|An,i(ba)||bn2k=kn(a)+1kn(b)bk,n2(ba)|+bn2(k=kn(a)+1kn(b)bk,n2)1/2×(k=kn(a)+1kn(b)(bk+i,nbk,n)2)1/2.

The last term in the right-hand side is going to zero as n , by taking into account condition (12.9). To show that the first term is going to zero as n , we observe that since we have proved that the triangular array (Xj,n)jZ satisfies the Lindeberg condition, we have supiZE(Xi,n2)0 . This implies that for any t ∈ [0, 1], ikn(t)E[Xi,n2]=bn2k=kn(t)bk,n2t . This proves that the first term in the right-hand side is going to zero as n . This ends the proof of (12.31) and then of (12.30).

12.5.2 The Case of Long-Range Dependent Statistics of Stationary Perturbed Determinantal Point Processes

We start by proving the asymptotic normality of linear statistics when the innovation process is a stationary perturbed determinantal point process.

(p.369) Corollary 12.22 Let X=(Xi,iZ) be a stationary determinantal point process and let Y=(Yi,iZ) be a non-degenerate stationary Gaussian sequence with positive continuous spectral density. Assume that X and Y are independent. Also, consider a measurable function h(x, y) defined on {0,1}×R and such that E[h2(X1,Y1)]< . Define the stationary sequence ξ=(ξi,iZ) by ξi=h(Xi,Yi)E[h(Xi,Yi)] , iZ . Let {bk,n,kZ} be a triangular array of non-negative numerical constants satisfying (12.9). Then ξ‎ has a continuous spectral density f on (−π‎, π‎] and

bn1kZbk,nξkd2πf(0)N,
(12.32)

where bn2=kZbk,n2 and NN(0,1) .

Proof of Corollary 12.22 By (10.3), recall that

ξi=XiυiXiψi+ζi,

where υ‎i = (h(1, Yi)−h(0, Yi))+, ψ‎i = (−h(1, Yi)+h(0, Yi))+ and ζ‎i = h(0, Yi). For any u=(u1,u2,u3)R3 , let

fu(x,y)=u1x(h(1,y)h(0,y))++u2x(h(1,y)+h(0,y))++u3h(0,y)

and define the process Uu:=(Ui(u))iZ by

Ui(u)=fu(Xi,Yi)E(fu(Xi,Yi))for anyiZ.
(12.33)

The fact that ξ‎ has a continuous spectral density f on (−π‎, π‎] comes from Lemma 12.23 (the proof of which follows) by applying it with u = (1, −1, 1).

Lemma 12.23 Let X=(Xi,iZ) , Y=(Yi,iZ) and h(⋅, ⋅) be as in Corollary 12.22. Then, for u=(u1,u2,u3)R3 , the stationary process U=(Ui(u),iZ) defined by (12.33) has a continuous spectral density fu on (−π‎, π‎].

Let us prove now the convergence in distribution of bn1kZbk,nξk . With this aim, we first recall that, for any uR+3 , limn0rn(Uu)=0 (see (10.4)). This convergence together with Lemma 12.23 and Corollary 12.21 prove that, for any uR+3 ,

bn1kZbk,nUk(u)d2πfu(0)N,
(12.34)

where fu is the spectral density of Uu. Now, by using the same arguments as those developed in the proof of Theorem 9.18 (and in particular Property 9.19), it follows that the convergence (12.34) also holds for any uR3 . Therefore taking u = (1, −1, 1), the convergence (12.32) follows. To end the proof of the corollary, it remains to prove Lemma 12.23.

(p.370) Proof of Lemma 12.23 The lemma will follow from Theorem 1.7 if one can prove that, for any uR3 ,

limnζn(U)=0whereζn(U):=supEiQUi(u)jSUj(u)Card(QS),
(12.35)

the supremum being taken over all pairs of non-empty, finite, disjoint sets Q, S Z satisfying d(Q,S)=minqQ,sS|qs|n .

With this aim, we first notice that since X=(Xi)iZ and Y=(Yi)iZ are independent,

EiQUi(u)jSUj(u)=ECovYiQfu(Xi,Yi),jSfu(Xj,Yj)+CovEYiQfu(Xi,Yi),EYjSfu(Xj,Yj).
(12.36)

But, setting y=(yi)iQS ,

CovY=yiQfu(Xi,Yi),jSfu(Xj,Yj)=CoviQb(yi)Xi,jSb(yj)Xj,

where b(yi) = u1(h(1, yi)−h(0, yi))+ + u2(−h(1, yi)+h(0, yi))+. With definition (1.17), we derive that, for any pairs (Q, S) of non-empty, finite, disjoint sets in Z satisfying d(Q, S) ≥ n,

CovY=yiQfu(Xi,Yi),jSfu(Xj,Yj)κn(X)kQ|b(yk)|21/2kS|b(yk)|21/2,

implying that

|E(covY(iQfu(Xi,Yi),jSfu(Xj,Yj))|κn(X)((kQE|b(Yk)|2)1/2(kSE|b(Yk)|2)1/2). 

Therefore, by the stationarity of Y,

ECovYiQfu(Xi,Yi),jSfu(Xj,Yj)κn(X)E|b(Y0)|2Card(QS).
(12.37)

(p.371) On the other hand, note that, by the stationarity of X,

E(fu(Xi,y))=u1E(X0)(h(1,y)h(0,y))++u2E(X0)(h(1,y)+h(0,y))++u3h(0,y):=v(y).

Therefore

CovEYiQfu(Xi,Yi),EYjSfu(Xj,Yj)=CoviQv(Yi),jSv(Yj)),

implying that, for any pairs (Q, S) of non-empty, finite, disjoint sets of Z satisfying d(Q, S) ≥ n,

CovEYiQfu(Xi,Yi),EYjSfu(Xj,Yj)ρn*(Y)iQv(Yi)2jSv(Yj)2.

Since Y has a positive and continuous spectral density, according to Fact 10.2, ρ1*(Y)<1 and then r1(Y)<1 which implies by Lemma 7.9 and stationarity that

iQv(Yi)22cCard(Q)v(Y0)22,

where c=1+r1(Y)1r1(Y)< . Therefore,

CovEYiQfu(Xi,Yi),EYjSfu(Xj,Yj)cρn*(Y)E|v(Y0)|2Card(QS).
(12.38)

So, overall, starting from (12.36) and taking into account (12.37) and (12.38), it follows that there exists a positive finite constant K such that, for any n ≥ 1,

ζn(U)K(κn(X)+ρn*(Y)).
(12.39)

By Fact 10.1, X is negatively dependent. Hence, by taking into account Remark 9.25, it follows that

k1|Cov(X0,Xk)|=k1Cov(X0,Xk)<,

(p.372) and then X has a continuous spectral density. Therefore, by Theorem 1.7, κn(X)0 as n . On the other hand, since Y is a Gaussian process with positive continuous spectral density, by Fact 10.2, ρn*(Y)0 as n . Taking into account these considerations in (12.39), it follows that (12.35) holds. This ends the proof of the lemma (and then the proof of Corollary 12.22).

Corollary 12.22 gives the following central limit theorem for linear processes constructed with non-negative numerical constants and generated by asymptotically negative dependent innovations.

Corollary 12.24 Suppose X=(Xk)kZ and Y=(Yk)kZ satisfy the condition of Corollary 12.22. Let h(x, y) be a measurable function defined on {0,1}×R and such that E[h2(X1,Y1)]< . Define the stationary sequence ξ=(ξi,Z) by ξi=h(Xi,Yi)E[h(Xi,Yi)] , iZ . Let {aj,jZ} be a non-negative sequence such that jZaj2< . Let

Uk=jZajξkj,Sn=k=1nUk,bk,n=i=1nak+iandbn2=kZbk,n2.

If bn then

bn1Snd2πf(0)N.

where NN(0,1) and f is the spectral density of Z.

Proof of Corollary 12.24 It suffices to notice that Sn=kZbk,nξk and to apply Corollary 12.22. Indeed since jZaj2< , bn2< . On another hand kZ(bk,nbk1,n)2jZaj2 which shows that the second part of condition (12.9) holds since bn2 and jZaj2< .

Comment The central limit theorem for triangular arrays of linear statistics of the following form

j=1kajiLjXi,

where Lj:=Lj(n) are sets and (Xi)iZ is a determinantal point process, was established in Soshnikov (2002). It is an open question whether the central limit theorem holds for long-range dependent statistics of perturbed determinantal processes for arbitrary sequences (aj) with aj2< .

12.6 Discrete Fourier Transform and Periodogram

An important application of martingale theory is in the analysis of periodogram. Given a stochastic process (Xj)jZ , the periodogram is defined as

(p.373)

In(θ)=12πnj=1nXjeijθ2,θR,

where i=1 is the imaginary unit. The periodogram is related to the discrete Fourier transform

Sn(θ)=j=1nXjeijθ.

In this section we consider a strictly stationary ergodic sequence (Xk)kZ of centered real-valued random variables with finite second moments, adapted to a non-decreasing filtration (Fk)kZ , and we shall impose the following regularity condition:

E(X0|F)=0P-almost surely,
(12.40)

where F=nZFn is the tail sigma field.

12.6.1 A CLT for Almost All Frequencies

We present below a central limit theorem for almost all frequencies that has been obtained by Peligrad and Wu in 2010. In Theorem 12.25, we let the parameter θ‎ be in the space [0, 2π‎], endowed with Borel sigma algebra and Lebesgue measure λ‎.

Theorem 12.25 Let (Xk)kZ be a strictly stationary ergodic sequence of centered real-valued random variables with finite second moments, adapted to a non-decreasing filtration (Fk)kZ , and such that (12.40) is satisfied. Then, for almost all θ‎ ∈ [0, 2π‎], the following convergence holds:

limnE|Sn(θ)|2n=2πf(θ),
(12.41)

where f is the spectral density of (Xk)kZ . Furthermore,

1n[Re(Sn(θ)),Im(Sn(θ))][N1(θ),N2(θ)]underP,
(12.42)

where N1(θ‎) and N2(θ‎) are independent identically distributed normal random variables with mean 0 and variance π‎f(θ‎).

An implication of this results is that we obtain the limiting distribution of the perio- dogram as follows:

Remark 12.26 As a consequence ofTheorem 12.25, for sequences satisfying (12.40), the perio- dogram n−1|Sn(θ‎)|2 is asymptotically distributed as f(θ‎)χ‎2(2) for almost all θ‎ ∈ [0, 2π‎], where χ‎2(2) is the chi-squared distribution with 2 degrees of freedom.

(p.374) The proof of Theorem 12.25 is a combination of martingale techniques with results from ergodic theory and harmonic analysis.

Some preparatory materials. From harmonic analysis we shall use the first three facts below. The last one comes from ergodic theory.

Fact 1 Carleson Theorem (Carleson 1966): If (ak)k≥0 are real numbers such that l=0al2<, then j=1najeijθ converges λ‎-almost surely on [0, 2π‎].

Fact 2 Hunt and Young (1974): There is a constant C such that

02πmax1kn=1keiθa2dθC=1na2.

Fact 3 Fejér–Lebesgue Theorem (cf Bary, 1964, p. 139 or Theorem 15.7 in Champeney, 1989). If g(θ‎) is integrable on [0, 2π‎], with Fourier coefficients

b=02πeiθg(θ)dθ

then, denoting

hm(θ)=12π=mmeiθb,

we have

g(θ)=limn1nm=nnhm(θ)λ-almost surely and inL1[0,2π].
(12.43)

Fact 4 Next proposition gives the strong law of large numbers for discrete Fourier transform. We notice that we need only the variables to be identically distributed to get an almost sure result. We refer the reader to Proposition 30 and Lemma 32 in Cuny, Merlevède and Peligrad (2013) and to Zhang (2017) for more general results on this type.

In the sequel λ‎ denotes the Lebesgue measure on [0, 2π‎].

Proposition 12.27 Let (Xk)k≥0 denote a sequence of identically distributed random variables on (Ω,K,P), with finite first moment. Then, for λ‎-almost all θ‎ in [0, 2π‎],

limn1nSn(θ)=0P-a.s.

where Sn(θ)=k=1neikθXk .

(p.375)

Proof We shall use a truncation argument. Define

Yk=XkI(|Xk|k)andSn*(θ)=k=1neikθYk.

Then,

k=1P(XkYk)=k=1P(|Xk|>k)E(|X0|)<.

By the Borel–Cantelli lemma, applied on Ω‎, P(XnYn i.o.) = 0. Therefore, for all θ‎ in [0, 2π‎],

1nSn(θ)and1nSn*(θ)have the sameP-a.s. limit.

The proposition is then reduced to show that, for almost all θ‎ in [0, 2π‎], we have that Sn*(θ)/n 0, P -a.s.

In order to prove it, note that, by using the Kronecker lemma, it is enough to show that, for almost all θ‎ in [0, 2π‎],

k=1eikθYkkconvergesP-a.s.
(12.44)

To prove it, we use Carleson’s (1966) theorem. Clearly,

k=1E(Yk2)k2=E(X02k=1I(|X0|k)k2)=E(X02k|X0|1k2)cE(|X0|),

for some positive constant c. The latter relation implies that there is ΩΩ with P(Ω)=1, such that, for all ωΩ ,

k=1Yk2(ω)k2<.

By Carleson’s (1966) theorem, for such ω‎,

k=1eikθYk(ω)kconverges forλ-almost allθ.

Denote now

A=(θ,ω)[0,2π]×Ω;wherek=1eikθYk(ω)kis convergent.

(p.376) By Fubini’s theorem, in the product space [0, 2π‎] ×Ω‎, we have λ×P(A)=1 . Again by Fubini’s theorem, (12.44) follows. The proof is complete.

Some preparatory lemmas. For kZ we define the projection operator by

Pk()=E(|Fk)E(|Fk1).
(12.45)

Lemma 12.28 Let

Tn(θ)=j=0nXjeijθ=X0+Sn(θ).

Under (12.40), for λ‎-almost all θ‎, we have

P0(Tn(θ))l=0eilθP0(Xl)=:D0(θ)P-almost surely and inL2.
(12.46)

Proof By (12.40) we have kZPk(X0)22=X022< , whence kZ|P0(Xk)|2< P -almost surely. Therefore, by Carleson’s (1966) theorem, for almost all ω‎, 1kneikθP0(Xk) converges λ‎-almost surely. Denote the limit by D0 = D0(θ‎). We now consider the set

A={(θ,ω)[0,2π]×Ω,where{P0(Sn(θ))}ndoes not converge}

and notice that almost all sections for ω‎ fixed have Lebesgue measure 0. So by Fubini’s theorem the set A has measure 0 in the product space and therefore, again by Fubini’s theorem, almost all sections for θ‎ fixed have probability 0. It follows that for almost all θ‎, P0(Sn(θ))D0 almost surely under P . Next, by the maximal inequality in Hunt and Young (1974), there is a constant C such that

02πsupn|P0(Sn(θ))|2λ(dθ)Ck|P0(Xk)|22

and then we integrate

02πEsupn|P0(Sn(θ))|2λ(dθ)CX022<.

Therefore,

Esupn|P0(Sn(θ))|2<for almost allθ.

Since |P0(Sn(θ))|<supn|P0(Sn(θ))|, and the last one is integrable for almost all θ‎, by the Lebesgue dominated convergence we have that P0(Sn(θ‎)) converges in L2 .

(p.377) Lemma 12.29 Let g(θ)=(2π)1E|D0(θ)|2 . For all jZ , we have

02πg(θ)eijθdθ=cj,
(12.47)

where cj = cov(X0, Xj) for all jZ . So (cj)jZ are the Fourier coefficients of g implying that g = f. Additionally, for almost all θ‎,

limnE|Sn(θ)|2n=2πf(θ).
(12.48)

Proof As before, let

P0(Tn(θ))=l=0neilθP0(Xl).

It follows that

12π02π|P0(Tn(θ))|2eijθdθ=12πu=0nv=0n02πP0(Xu)P0(Xv)ei(uv+j)θdθ.

Since

12π02πeijθdθ=0ifj0,

for 0 ≤ jn,

12π02π|P0(Tn(θ))|2eijθdθ=l=0njP0(Xl)P0(Xl+j),
(12.49)

and 0 in rest (i.e. if j < 0 or j > n). Since Xj=lZPl(Xj) , by orthogonality of martingale differences and stationarity we have that

cj=limNEl=N0Pl(X0)l=N0Pl(Xj)=limNl=N0E[Pl(X0)Pl(Xj)]=limnl=0njE[(P0Xl)(P0Xl+j)].

Combining the latter equality with (12.49) we obtain

cj=limn12π02πE|P0(Tn(θ))|2eijθdθ.

(p.378) We know that, by (12.46), for λ‎-almost all θ‎,

P0(Tn(θ))D0(θ)P-almost surely and inL2.

By the Lebesgue dominated convergence theorem, as in the proof of Lemma 12.28, (12.47) follows in view of the Hunt and Young maximal inequality since supn|P0(Tn(θ))| is integrable.

Now we prove (12.48). By stationarity, we have

1nE|Sn(θ)|2=1nj=1nl=1nE(XjXl)eijθeilθ=1nj=1n1l=1ncjlei(jl)θ=j=(n1)n11|j|ncjeijθ.

Namely E|Sn(θ)|2/n is the Cesàro average of the sum j=llcjeijθ . Note that D0(θ)22 is integrable over [0, 2π‎]. Therefore by the Fejér–Lebesgue Theorem (Fact 3 above), relation (12.48) holds for λ‎-almost all θ‎ ∈ [0, 2π‎].

Proof of Theorem 12.25 The first assertion of Theorem 12.25 is just Lemma 12.29. We now prove (12.42).

Step 1. The construction of a martingale differences sequence. Define the projector operator by (12.45). Then we construct

P1(Sn(θ))=E(Sn(θ)|F1)E(Sn(θ)|F0)=k=1neikθP1(Xk).

Then, by Lemma 12.28 for λ‎-almost all θ‎,

P1(Sn(θ))D1(θ)inL2.
(12.50)

Note that

E(P1(Sn(θ))|F0)=0almost surely underP.

Therefore, by the contractive property of the conditional expectation,

0=E(P1(Sn(θ))|F0)E(D1(θ)|F0)inL2.

For λ‎-almost all θ‎, we then construct a sequence of stationary martingale differences (Dk(θ‎))k≥1, given by

Pk(Sn+k(θ)Sk(θ))Dk(θ)inL2.

(p.379) Step 2. Martingale approximation. By step 1 we know that there is a set Γ1[0,2π] , such that λ(Γ1)=2π and for all θΓ1 the martingale

Mn(θ)=1kneikθDk(θ)

is well defined in L2 . We show now that, for almost all θ‎,

E|Sn(θ)Mn(θ)|2n0.
(12.51)

To this end, note that Sn(θ)E(Sn(θ)|F0) and E(Sn(θ)|F0) are orthogonal, and we have

Sn(θ)22=Sn(θ)E(Sn(θ)|F0)22+E(Sn(θ)|F0)22.
(12.52)

Next, for θΓ1 , by the orthogonality of martingale differences, the stationarity and Lemma 12.28, we have that

Sn(θ)E(Sn(θ)|F0)Mn(θ)22=k=1nPk(Sn(θ)Mn(θ))22=k=1nPkSn(θ)eikθDk(θ)22=k=1nP0(Tnk(θ))D0(θ)22=o(n).
(12.53)

Hence,

limn|Sn(θ)E(Sn(θ)|F0)2Mn(θ)2|n=0.
(12.54)

But Mn(θ)22=nD0(θ)22 and by Lemma 12.37, for almost all θ‎, limnn1Sn(θ)22=D0(θ)22 . This consideration together with (12.52) and (12.54) entail that, for almost all θ‎,

limsupnE(Sn(θ)|F0)22n=0.

This implies (12.51) by taking into account (12.53) and the decomposition

Sn(θ)Mn(θ)22=Sn(θ)E(Sn(θ)|F0)Mn(θ)22+E(Sn(θ)|F0)22.

(p.380) Step 3. The CLT for the approximating martingale. Here we shall consider θ‎ in a set of measure 2π‎ where all the functions make sense.

By Theorem 1.9, it remains just to prove the central limit theorem for complex valued martingale:

1n1kneikθDk(θ).

As a matter of fact we shall provide a central limit theorem for the real part and imaginary part and show that in the limit they are independent. By the Cramér–Wold device, we then only have to show that, for any reals a and b,

1n1kn(aRe(eikθDk(θ))+bIm(eikθDk(θ)))dN~N(0,π(a2+b2)f(θ)),as n.

With this aim, we shall apply Theorem 2.29. Clearly, since D0(θ‎) is square integrable and (Dk(θ))kZ is stationary,

1nE(max1kn|aRe(eikθDk(θ))+bIm(eikθDk(θ))|)|a|+|b|nE(max1kn|Dk(θ)|)0.

It remains to verify

1n1knaRe(eikθDk(θ))+bIm(eikθDk(θ))2P(a2+b2)2E|D0(θ)|2,

or equivalently if a2 + b2 = 1,

1n1knaRe(eikθDk(θ))+bIm(eikθDk(θ))2P12E|D0(θ)|2.
(12.55)

For convenience we write Dk = Dk(θ‎) and Dk = Ak + iBk . So,

(aRe(eikθDk(θ))+bIm(eikθDk(θ)))2=(acos(kθ)+bsin(kθ))2Ak2+(bcos(kθ)asin(kθ))2Bk2+2(acos(kθ)+bsin(kθ))(bcos(kθ)asin(kθ))AkBk.

By using basic trigonometric formulas, it follows that, if a2 + b2 = 1,

(aRe(eikθDk(θ))+bIm(eikθDk(θ)))2=Ak2+Bk22+(a2b2)cos(2kθ)2(Ak2Bk2)+absin(2kθ)(Ak2Bk2)+ab(cos(2kθ)+sin(2kθ))AkBk+(b2a2)sin(2kθ)AkBk.

(p.381) By stationarity and the ergodic theorem,

1n1knAk2+Bk22=12n1kn|Dk(θ)|212E|D0(θ)|2P-almost surely

On another hand, by Proposition 12.27, for almost all θ‎,

(aRe(eikθDk(θ))+bIm(eikθDk(θ)))2Ak2+Bk220P-almost surely.

By these arguments, (12.55) follows and it remains to apply Lemma 12.29 to obtain the result.

12.6.2 Examples

We present several examples of processes for which the conclusions of Theorem 12.29 hold.

Clearly condition (12.40) is satisfied if the left tail sigma field F is trivial. These processes are called regular (see Chapter 2, vol. 1 in Bradley, 2007).

Example 1 (Mixing sequences.)

Assume that (Xk)kZ is a strictly stationary sequence of real-valued r.v.’s. If the sequence is additionally strong mixing (so limnα(n)=0 where the strong mixing coefficients α‎(n) have been defined in Section 5.1), the tail sigma field is trivial; see Claim 2.17a in Bradley (2007). Examples of this type include Harris recurrent Markov chains. Now if we denote by (ρ‎(n))n≥1 the sequence of maximal correlation coefficients associated with (Xk)kZ (see again Section 5.1 for the definition), and if limnρ(n)<1 , then the tail sigma field is also trivial; see Proposition 5.6 in Bradley (2007).

Example 2 (Functions of i.i.d. random variables.)

Let (εk)kZ be a sequence of i.i.d. and consider Xn = f(ε‎k, kn). These are regular processes and therefore Theorem 12.25 applies. Examples include linear processes, functions of linear processes and iterated random functions (Wu and Woodroofe, 2004) among others. For example let Xn=j=0ajεnj , where the ε‎j’s are i.i.d. with mean 0 and variance 1 and aj are real coefficients with j=1aj2< . In this case Xn is well-defined and, by Lemmas 12.28 and 12.29, the spectral density is

f(θ)=12πj=0ajeijθ2.

Example 3 (Reversible Markov chains.)

As in Chapter 14, we consider (Xj)jZ a strictly stationary, centered, ergodic and reversible Markov chain with values in a measurable space. By computation (14.4),

E(X0|Fn)22=cov(X0,X2n):=c2n.

If we assume cn0 then the conclusion of Theorem 12.25 holds.