asymptotics of spectral density estimates - the

Econometric Theory, 26, 2010, 1218–1245.doi:10.1017/S026646660999051X

ASYMPTOTICS OF SPECTRALDENSITY ESTIMATES

WEIDONG LIUZhejiang University

WEI BIAO WUUniversity of Chicago

We consider nonparametric estimation of spectral densities of stationary processes,a fundamental problem in spectral analysis of time series. Under natural and easilyverifiable conditions, we obtain consistency and asymptotic normality of spectraldensity estimates. Asymptotic distribution of maximum deviations of the spectraldensity estimates is also derived. The latter result sheds new light on the classicalproblem of tests of white noises.

1. INTRODUCTION

A fundamental problem in spectral analysis of time series is the estimation ofspectral density functions. Let Xk,k ∈ Z, be a stationary process with mean zeroand finite covariance function γk = E(X0 Xk). Assume that

∑k∈Z

|γk | < ∞. (1.1)

Let ı = √−1 denote the imaginary unit. Under (1.1), the spectral density function

f (θ) = 1

2π∑k∈Z

γkeıkθ = 1

2π∑k∈Z

γk cos(kθ), 0 ≤ θ < 2π, (1.2)

exists and is continuous. The primary goal of the paper is to consider asymptoticproperties of estimates of f . Based on observations X1, . . . , Xn , let the samplecovariances

γk = 1

n

n

∑i=|k|+1

Xi Xi−|k|, 1−n ≤ k ≤ n −1. (1.3)

It is well known that the periodogram

In(θ) = 1

2πn|Sn(θ)|2 = 1

2πn

n−1

∑k=1−n

γkeıkθ , where Sn(θ) =n

∑k=1

Xkeıkθ ,

We gratefully acknowledge helpful comments from a co-editor and from three anonymous referees that led to a muchimproved version. Address correspondence to Weidong Liu, Department of Mathematics, Zhejiang University,Huangzhou, Zhejiang, China; e-mail: [email protected].

1218 c© Cambridge University Press 2009 0266-4666/10 $15.00

ASYMPTOTICS OF SPECTRAL DENSITY ESTIMATES 1219

is an asymptotically unbiased but inconsistent estimate of f (θ). In the paper weconsider the lag-window estimate

fn(θ) = 1

2π

n−1

∑k=1−n

K (k/Bn)γkeıkθ , (1.4)

where bn = B−1n is the bandwidth satisfying bn → 0 and nbn → ∞ and the kernel

K is symmetric and bounded, K (0) = 1, and K is continuous at zero. If K hasbounded support, because nbn → ∞, the summands for large k in (1.4) are zero.The function K (·/Bn) becomes more concentrated at the origin for bigger Bn .Let

γk = 1

n

n

∑i=|k|+1

(Xi − Xn)(Xi−|k| − Xn), where Xn = ∑ni=1 Xi

n.

A variant of (1.4) that allows unknown mean μ = EX j is the following esti-mate:

fn(θ) = 1

2π

n−1

∑k=1−n

K (k/Bn)γkeıkθ . (1.5)

Properties of spectral density estimates have been explored in many classicaltextbooks on time series; see, for example, Anderson (1971), Brillinger (1975),Brockwell and Davis (1991), Grenander and Rosenblatt (1957), Priestley (1981),and Rosenblatt (1985), among others. See Shao and Wu (2007) for further ref-erences. It seems that many of the previous results require restrictive conditionson the underlying processes such as linear processes or strong mixing processes.Other contributions can be found in Phillips, Sun, and Jin (2006, 2007) andVelasco and Robinson (2001). In this paper we shall present an asymptotic theoryfor fn(θ) under very mild and natural conditions, thus substantially extendingthe applicability of spectral analysis to nonlinear and/or non–strong mixing pro-cesses. Some open problems are solved under our dependence framework (2.1).

The rest of the paper is structured as follows. Main results are presented inSection 2 and proved in the Appendix. Section 3 provides bounds for approxi-mations by m-dependent random variables, and Section 4 presents inequalitiesfor m-dependent processes. Section 5 proves a very general central limit theorem(CLT) for quadratic forms of stationary processes that is of independent interest.Many classical results are special cases of Theorem 6 in Section 5.

We now introduce some notation. We say that a random variable X ∈Lp, p > 0,if ‖X‖p := [E(|X |p)]1/p < ∞. Write ‖·‖ = ‖·‖2. For u,v ∈R, let u = max{k ∈Z : k ≤ u}, u ∨v = max(u,v), and u ∧v = min(u,v). Let C be the set of complexnumbers. Denote by Cp a constant that only depends on p and denote by C anabsolute constant. Their values may vary from display to display. For two positivesequences (an) and (bn), write an ∼ bn if limn→∞ an/bn = 1 and an � bn if, forsome c > 0, c ≤ an/bn ≤ c−1 holds for all sufficiently large n. Define ω(u) = 2 ifu/π ∈ Z and ω(u) = 1 if u/π �∈ Z.

1220 WEIDONG LIU AND WEI BIAO WU

2. MAIN RESULTS

Assume throughout the paper that εj , j ∈ Z, are independent and identically dis-tributed (i.i.d.) random variables and R is a measurable function such that

X j = R(. . . ,εj−1,εj ) = R(Fj ), where Fj = (. . . ,εj−1,εj ), (2.1)

is well defined. The class of processes under the framework (2.1) is huge; seeWiener (1958), Priestley (1988), Tong (1990), and Wu (2005), among others. Shaoand Wu (2007) provide examples of nonlinear time series that are of form (2.1).

To develop an asymptotic theory for the spectral density estimate fn(·), we needto introduce appropriate dependence measures. Following Wu (2005), we shallapply the idea of coupling and use a physical dependence measure. Let {εj ,ε

′k :

j,k ∈ Z} be i.i.d. random variables. For a set T ⊂ Z, let εj,T = ε′j if j ∈ T , and

εj,T = εj if j �∈ T . Let Fj,k,T = (εl,T , j ≤ l ≤ k). If a random variable W is afunction of F−∞,∞, say, W = w(F−∞,∞), write WT = w(F−∞,∞,T ). For X j ∈Lp, p > 0, and T = {0}, define the physical dependence measure

δj,p = ‖X j − X j,{0}‖p. (2.2)

Here, by our convention, X j,{0} = R(Fj,{0}), where Fj,{0} = (. . . ,ε−1,ε′0,ε1, . . . ,

εj ). Namely, X j,{0} is obtained by replacing ε0 in X j by ε′0. If j < 0, δj,p = 0.

If we view Fj as input and X j as output of a physical system, then δj,p mea-sures the dependence of X j on the input ε0 via coupling. In many situations it iseasy to work with δj,p, which is directly related to the underlying data-generatingmechanism (Wu, 2005).

Example 1

Let X j = g(∑∞l=0 alεj−l), where al are real coefficients, εl are i.i.d. with εl ∈ Lp,

p > 0, and g is a Lipschitz continuous function. For j ≥ 0, we have X j,{0} =g(∑ j−1

l=0 alεj−l + ajε′0 + ∑∞

l=1+ j alεj−l). Hence δj,p = ‖X j − X j,{0}‖p = O(|aj |‖ε0 − ε′

0‖p) = O(|aj |). In the special case in which g(x) = x , al = 2−l and εl

are i.i.d. with P(εl = 1) = P(εl = −1) = 12 , the process (X j ) is not strong mixing

(Andrews, 1984).

Example 2

Let (X j ) be a nonlinear time series recursively defined by X j = g(X j−1,εj ),where εj are i.i.d. and g is a measurable function. Assume that there exist p > 0and x0 ∈ R such that g(x0,ε0) ∈ Lp and E(L p

ε0) < 1, where Lε0 = supx �=x ′ |g(x,ε0) − g(x ′,ε0)|/|x − x ′|. Then (X j ) has a unique stationary solution of theform (2.1), and δj,p = O(ρ j ) for some ρ ∈ (0,1) (Wu, 2005). Shao and Wu (2007)showed that the latter holds for a variety of processes including autoregressive–autoregressive conditionally heteroskedastic processes, amplitude-dependentexponential autoregressive processes, asymmetric generalized autoregressive con-ditionally heteroskedastic processes, and signed volatility models.


Sections 2.1, 2.2, and 2.3 concern consistency, asymptotic normality, andmaximum deviations of fn(·), respectively. Our results are all based on δj,p.Define

m,p =∞∑

j=mδj,p and m,p =

( ∞∑

j=mδ

p′j,p

)1/p′

, where p′ = min(2, p). (2.3)

Theorems 1 and 2 in Sections 2.1 and 2.2 require the short-range dependencecondition 0,p < ∞; namely, the cumulative dependence of ε0 on the futurevalues (X j )j≥0 is finite. A careful check of the proofs of Theorems 1–6 indi-cates that analogous results also hold for the two-sided process X j = R(. . . ,εj−1,εj ,εj+1, . . .) because our main tool is the m-dependence approximation; seeSection 3. For two-sided processes, similar approximations hold. Details are omit-ted because this does not involve essential extra difficulties.

2.1. Consistency

To state our consistency result, we need some regularity conditions on the kernelK . Slightly different forms are needed for asymptotic normality and maximumdeviations; see Conditions 2 and 3 in Sections 2.2 and 2.3. All those conditionson K are mild, and they are satisfied for Parzen, triangle, Tukey, and many othercommonly used windows (Priestley, 1988).

Condition 1. Assume that K is a bounded, absolutely integrable, even function,K (0) = 1 and K is continuous. Further assume that limw→0 w∑k∈Z K 2(kw) =∫∞−∞ K 2(u)du =: κ < ∞ and its Fourier transform K (x) = ∫∞

−∞ K (u)eıxu du

satisfies∫∞−∞ |K (x)|dx < ∞.

THEOREM 1. Let Condition 1 be satisfied. Assume that E(Xk) = 0, Xk ∈ Lp,p ≥ 2, and 0,p = ∑∞

j=0 δj,p < ∞. Let Bn → ∞ and Bn = o(n) as n → ∞. Then

supθ∈R

‖ fn(θ)− f (θ)‖p/2 → 0. (2.4)

Because fn and f are even and have period 2π , supθ∈R in (2.4) is equivalentto supθ∈[0,π ].

Remark 1. By Theorem 2 in Wu (2005), under 0,p < ∞, we have ‖X1 + ·· ·+ X j‖p = O(

√j ). Hence, for 0 ≤ k ≤ n − 1, ‖Xn ∑n

i=1+k Xi−k‖p/2 ≤ ‖Xn‖p

‖∑ni=1+k Xi−k‖p = O(1), from which, by elementary calculations, we obtain

max|k|≤n−1 ‖γk − γk‖p/2 = O(n−1). Assume

n−1

∑k=1−n

|K (k/Bn)| = O(Bn). (2.5)


Then (2.4) also holds if fn therein is replaced by fn in view of∥∥∥∥supθ∈R

| fn(θ)− fn(θ)|∥∥∥∥

p/2

≤ 1

2π

n−1

∑k=1−n

|K (k/Bn)| max|k|≤n‖γk − γk‖p/2 = O(Bn/n).

(2.6)

With (2.6), Theorems 2–5 in Sections 2.2 and 2.3 also hold if fn therein is re-placed by fn . Condition (2.5) holds for Epanechnikov, triangle, Parzen, and manyother commonly seen kernels.

Theorem 1 imposes very mild conditions. Clearly we need Bn → ∞ and Bn =o(n) to ensure consistency. The short-range dependence condition 0,2 < ∞(Wu, 2005) implies (1.1) and hence entails the existence of the spectral densityfunction. If 0,2 = ∞, then f may not exist. Consider, for example, the linearprocess X j = ∑∞

l=0 alεj−l , where εl are i.i.d. with mean zero and variance 1. Thenδj,2 = |aj |‖ε0 − ε′

0‖ = |aj |√

2 and 2π f (θ) = |∑∞j=0 aj eı jθ |2. If ∑∞

j=0 aj = ∞, for

example, aj = j−β , j ∈N, β ∈ ( 12 ,1), then f has a pole at θ = 0, and the left-hand

side of (2.4) is ∞. In this case ∑k∈Z γk = ∞, and the process (X j ) is long-rangedependent.

Davidson and de Jong (2002) considered a closely related problem of estimat-ing the variance s2

n = var(X1 +·· ·+ Xn), in which Xi are mean zero random vari-ables. They proved that, for the process Xi = ∑∞

j=0 ajηi− j , where ∑∞j=0 |aj | < ∞

and ηn is L2-near epoch dependent (NED) of size − 12 , s2

n = ∑nk=1 X2

k + 2∑Bnk=1

K (k/Bn)nγk satisfies s2n/s2

n → 1 in probability. Their result and our Theorem 1have different ranges of applicability. Consider the case that both results are ap-plicable: Xi = ∑∞

j=0 ajηi− j , where ηi = (∑∞j=0 bjεi− j )

2 −∑∞j=0 b2

j and εi are i.i.d.

N (0,1). Their condition of L2 NED of size − 12 requires that ‖ηn −E(ηn|ε1, . . . ,

εn)‖ = O(n−q) for some q > 12 . With elementary calculations, the latter condition

is reduced to ∑∞j=n b2

j = O(n−2q), which implies that ∑∞j=2 |bj | ≤ ∑∞

k=1

(∑2k+1−1

j=2k

b2j

) 12 2k/2 = ∑∞

k=1 2k/2 O(2−qk) < ∞. We now apply our Theorem 1 with p = 2.Elementary calculations show that physical dependence measure δn,2 = ∑n

i=0O(|ai bn−i |). Hence our condition 0,2 < ∞ only requires ∑∞

j=0 |aj | < ∞ and∑∞

j=0 |bj | < ∞. Hence in this example their NED-based condition is slightlystronger. Jansson (2002) considered the consistency of covariance matrix estima-tion for linear process with ηn being Rd -valued martingale differences. In thespecial case d = 1, Jansson’s result requires Bn = o(

√n), whereas our result

permits Bn = o(n).

2.2. Asymptotic Normality

A classical problem in spectral analysis of time series is to develop an asymptoticdistributional theory for the spectral density estimate fn(θ). With the latter resultsone can perform statistical inference such as hypothesis testing and constructionof confidence intervals. However, it turns out that the central limit problem for


fn(θ) is highly nontrivial. Earlier results require stringent conditions. The case oflinear processes has been dealt with in Anderson (1971). Rosenblatt (1984) ob-tained a central limit theorem (CLT) under strong mixing and cumulant summabil-ity conditions. More restrictive cumulant conditions are used in Brillinger (1969).Bentkus and Rudzkis (1982) dealt with Gaussian processes. Shao and Wu (2007)required the condition that δi,p converges to zero geometrically fast.

Here we present a CLT for fn(θ) under very mild and natural conditions, andit allows a wide class of nonlinear processes. In Theorem 2, the condition ondependence 0,4 < ∞ is natural, because otherwise the process (X j ) may belong-range dependent and the spectral density function may not be well defined.In Rosenblatt (1985), a summability condition of eighth-order joint cumulants isrequired. Rosenblatt asked whether the eighth-order summability condition can beweakened to the fourth order. The latter conjecture is solved in Theorem 2 in thesense that it imposes a summability condition of fourth-order physical dependencemeasures, which in many applications has the additional advantage that it is easierto work with than conditions on joint cumulants. The bandwidth condition Bn →∞ and Bn = o(n) is also natural; see our consistency result, Theorem 1. Recallthat ω(u) = 2 if u/π ∈ Z and ω(u) = 1 if u/π �∈ Z. Theorem 2 is proved in theAppendix.

Condition 2. K is symmetric and bounded, limu→0 K (u) = K (0) = 1, andκ := ∫∞

−∞ K 2(x)dx < ∞. Further assume that K is continuous at all but a finitenumber of points and suppose that sup0<w≤1 w∑j≥c/w K 2( jw) → 0 as c → ∞.

THEOREM 2. Assume E(Xk) = 0, E(X4k ) < ∞, and 0,4 < ∞. Let Bn → ∞

and Bn = o(n) as n → ∞. Then under Condition 2, for any fixed 0 ≤ θ ≤ π ,√n

Bn{ fn(θ)−E[ fn(θ)]} ⇒ N [0,σ 2(θ)], where σ 2(θ) = ω(θ) f 2(θ)κ. (2.7)

Remark 2. Theorem 2 is applicable for a very wide range of bandwidths. Inpractice, one can use the bandwidth selector in Buhlmann and Kunsch (1999),Politis, Romano, and Wolf (1999), or Song and Schmeiser (1995).

Remark 3. The bias E[ fn(θ)] − f (θ) can be calculated by some standard ar-guments; see Anderson (1971) or Priestley (1981).

2.3. Maximum Deviations

Theorem 2 provides a CLT for fn(θ) −E[ fn(θ)]. In the inference of spectra,one often needs to know the asymptotic distribution of the maximum deviationsup0≤θ≤π | fn(θ) − f (θ)|. Such a result can be used to construct simultaneousconfidence bands for f (θ) over θ ∈ [0,π ] and to conduct a parametric specifica-tion test for f . For example, if a constant function can be embedded into the band,then we can accept the hypothesis that (Xk) is a white noise sequence. However,the maximum deviation problem is extremely difficult. In 1967, Woodroofe and


Van Ness considered linear processes and obtained an asymptotic theory for maxi-mum deviations. Rudzkis (1985) considered the special Gaussian processes. Overthe past 40 years, however, it seems that there has been no significant progress ongeneralizing their results to nonlinear processes.

Shao and Wu (2007) posed an open problem whether an asymptotic distribu-tional theory for maximum deviations can be obtained for a wide class of non-linear time series satisfying the geometric-moment contraction condition δn,p =O(ρn). Theorem 5 solves the conjecture by considering maxi≤Bn | fn(λ

∗i )−E fn

(λ∗i )|/ f (λ∗

i ), where λ∗i = π |i |/Bn . Theorems 3 and 4 present similar results under

weaker dependence conditions.

Condition 3. K is an even, bounded function with bounded support [−1,1],limu→0 K (u)= K (0)= 1, κ := ∫ 1

−1 K 2(u)du < ∞, and ∑j∈Z sup|s− j |≤1 |K ( jw)−K (sw)| = O(1) as w → 0.

Let the total variation V ba (K ) = sup{∑l

i=1 |K (xi )−K (xi−1)| : a = x0 < x1< · · ·< xl = b}. Then sup|s− j |≤1 |K ( jw)− K (sw)| ≤ V jw+w

jw−w (K ). If K has boundedvariation, namely, V ∞−∞(K )<∞, then ∑j∈Z sup|s− j |≤1 |K ( jw)− K (sw)| = O(1).Popular choices such as triangle, Epanechnikov, quartic, and other kernels all havebounded variations.

Condition 4. There exists 0 < δ < δ < 1 and c1,c2 > 0 such that, for all largen, c1nδ ≤ Bn ≤ c2nδ holds.

Condition 5.

(a) Let dm,q = ∑∞t=0 min(δt,q ,m+1,q). Assume dn,p = O(n−T1) with T1 >

max[

12 − (p −4)/(2pδ), 2δ/p

].

(b) n,p = O(n−T2), T2 > max[0, 1− (p −4)/(2pδ)].

THEOREM 3. Assume X0 ∈Lp, p > max(4,2/(1−δ)), and EX0 = 0. Furtherassume Conditions 3, 4, 5(a), and 5(b). Let λ∗

i = π |i |/Bn. Then, for all x ∈ R,

P

[max

0≤i≤Bn

n

Bn

| fn(λ∗i )−E fn(λ∗

i )|2f 2(λ∗

i )κ−2log Bn + log(π log Bn) ≤ x

]→ e−e−x/2

.

(2.8)

Theorem 3 requires the moment condition Xi ∈ Lp with p > max(4,2/(1 −δ)) → ∞ as δ → 1. Theorems 4 and 5 aim to weaken the latter moment condition.

THEOREM 4. Assume EX0 = 0, X0 ∈ Lp, p > 4, and Conditions 3, 4, and5(a). Further assume that K is continuous and K (x) := ∫∞

−∞ e−iλx K (λ)dλ satis-

fies∫∞−∞ |K (x)|dx < ∞. Then (2.8) holds.

THEOREM 5. Assume Conditions 3 and 4 and EX0 = 0, X0 ∈ Lp, p > 4.Further assume that δn,p = O(ρn) for some 0 < ρ < 1. Then (2.8) holds.


Remark 4. Theorems 3–5 allow nonlinear processes. When they are applied tolinear processes, our conditions are weaker than the classical one in Woodroofeand Van Ness (1967). To derive (2.8), the latter paper requires δ < 2

5 , δ ≥ 14 ,

ε0 ∈ L8, and |ak | = O(k−1−β) with β > 15 . If p = 8 and δ < 2

5 , our Theorem 4

only requires the weaker condition |ak | = O(k−1−β) with β > (√

14−2)/10. Wealso note that the requirement on β becomes weaker for smaller δ. Additionally,we allow smaller p with 4 < p < 8.

Remark 5. If K (x)−1 = O(x) as x → 0 and ∑k≥1 kδk,2 < ∞, then E fn(θ)−f (θ) = O(B−1

n ). Hence we can replace E fn(λ∗i ) in (2.8) by f (λ∗

i ) if n logn = o(B3

n ).

3. APPROXIMATIONS BY m-DEPENDENT PROCESSES

With the physical dependence measure δj,p in (2.2), we are able to provideexplicit error bounds for approximating functionals of Xk by functionals of them-dependent process

Xt := Xt,m = E(Xt |εt−m, . . . ,εt ) = E(Xt |Ft−m,t ), m ≥ 0, (3.1)

where Ft−m,t = σ(εt−m, . . . ,εt ). Define the projection operator Pk by

Pk · = E(·|Fk)−E(·|Fk−1), k ∈ Z.

Lemma 1, which follows, concerns linear forms, whereas Proposition 1 in thissection is for quadratic forms. Proposition 2 in this section gives a martingaleapproximation for quadratic forms of m-dependent processes.

LEMMA 1. Assume Xi ∈ Lp, p > 1, and E(Xk) = 0. Let Cp = 18p3/2(p −1)−1/2 and p′ = min(2, p). Let α1,α2, . . . ,∈ C. Then∥∥∥∥∥ n

∑k=1

αk(Xk − Xk)

∥∥∥∥∥p

≤ Cp Anm+1,p, where An =(

n

∑k=1

|αk |p′)1/p′

. (3.2)

Also, we have (i) ‖∑nk=1 αk Xk‖p ≤ Cp An0,p and (ii) ‖∑n

k=1 αk Xk‖p ≤ Cp An

0,p.

Proof. Let Dk, j :=E(Xk |Fk− j,k)−E(Xk |Fk− j+1,k). Then Dk, j , k = n, . . . ,1,form martingale differences with respect to Fk− j,∞, and ‖Dk, j‖p ≤ δj,p. ByMinkowski’s and Burkholder’s inequalities (see Wu and Shao, 2007, Lem. 1),we have∥∥∥∥∥ n

∑k=1

αk Dk, j

∥∥∥∥∥p′

p

≤ C p′p

n

∑k=1

‖αk Dk, j‖p′p ≤ C p′

p

n

∑k=1

|αk |p′δ

p′j,p.

Because Xk − Xk = ∑∞j=1+m Dk, j , (3.2) follows.


Because Xk = ∑∞j=0 Dk, j , the preceding argument implies (i). By Jensen’s in-

equality,

δk,p := ‖Xk − Xk,{0}‖p = ‖E(Xk |Fk−m,k)−E(Xk,{0}|Fk−m,k,{0})‖p

= ‖E(Xk − Xk,{0}|Fk−m,k,ε′0)‖p ≤ δk,p. (3.3)

So (ii) follows from (i). n

PROPOSITION 1. Assume EX0 = 0, E|X0|2p < ∞, p ≥ 2, and 0,2p < ∞.Let

Ln = ∑1≤ j< j ′≤n

αj ′− j X j X j ′ and Ln = ∑1≤ j< j ′≤n

αj ′− j X j X j ′ ,

where α1,α2, . . . ,∈ C and Xt is defined in (3.1). Let An = (∑n−1s=1 |αs |2)1/2. Then

‖Ln −ELn − (Ln −ELn)‖p

n12 An0,2p

≤Cpdm,2p, where dm,q =∞∑t=0

min(δt,q ,m+1,q).

(3.4)

Proof. Let Zt−1 = ∑t−1j=1 αt− j X j , Zt−1 = ∑t−1

j=1 αt− j X j , Yt = Xt Zt−1, Yt =Xt Zt−1, and

L�n = ∑

1≤ j< j ′≤n

αj ′− j X j X j ′ =n

∑t=2

Xt Zt−1.

Recall that the notation X j,{k} represents a coupled version of Xi = R(Fi ) byreplacing εk in Fi by ε′

k . If k > j , then X j,{k} = X j . We similarly define Zt−1,{k}and Zt−1,{k}. So

‖Pk(Ln − L�n)‖p ≤

∥∥∥∥∥ n

∑t=2

Xt (Zt−1 − Zt−1)− Xt,{k}(Zt−1,{k} − Zt−1,{k})∥∥∥∥∥

p

≤∥∥∥∥∥ n

∑t=2

Xt,{k}[(Zt−1 − Zt−1)− (Zt−1,{k} − Zt−1,{k})]∥∥∥∥∥

p

+n

∑t=2

‖(Xt − Xt,{k})(Zt−1 − Zt−1)‖p =: Ik + I Ik . (3.5)

By (3.3), ‖X j − X j,{k}‖2p ≤ δj−k,2p. Because ‖X j − X j‖2p ≤ m+1,2p, we have‖X j − X j − X j,{k} + X j,{k}‖2p ≤ 2min(δj−k,2p,m+1,2p). By Lemma 1,

Ik =∥∥∥∥∥ n

∑t=2

Xt,{k}t−1

∑j=1

αt− j (X j − X j − X j,{k} + X j,{k})∥∥∥∥∥

p


=∥∥∥∥∥n−1

∑j=1

(X j − X j − X j,{k} + X j,{k})n

∑t=1+ j

αt− j Xt,{k}

∥∥∥∥∥p

≤ 2n−1

∑j=1

min(δj−k,2p,m+1,2p)C2p An0,2p.

By Lemma 1, ‖Zt−1 − Zt−1‖2p ≤ C2p Anm+1,2p. Because ∑nt=2 δt−k,2p ≤ 0,2p

and ‖(Xt − Xt,{k})(Zt−1 − Zt−1)‖p ≤ ‖Xt − Xt,{k}‖2p‖Zt−1 − Zt−1‖2p,

n

∑k=−∞

I I 2k ≤ Cp A2

n2m+1,2p

n

∑k=−∞

0,2p

n−1

∑j=1

δj−k,2p ≤ Cp A2nn2

m+1,2p20,2p,

n

∑k=−∞

I 2k ≤ Cp A2

n20,2p

n

∑k=−∞

0,2p

n−1

∑j=1

min(δj−k,2p,m+1,2p)

≤ Cp A2n2

0,2pn0,2pdm,2p.

Hence, by (3.5), because m+1,2p ≤ dm,2p,

‖Ln −ELn − (L�n −EL�

n)‖2p ≤ Cp

n

∑k=−∞

‖Pk(Ln − L�n)‖2

p ≤ Cpn A2n

20,2pd2

m,2p,

which, by a similar inequality for ‖L�n −EL�

n − (Ln −ELn)‖2p, implies (3.4). n

PROPOSITION 2. Assume EX0 = 0, X0 ∈ L4, and 0,4 < ∞. Let αj =βj eı jλ, where λ ∈R, βj ∈R, 1−n ≤ j ≤ −1, m ∈N, and Ln = ∑1≤ j<t≤n αj−t X j

Xt . Define

Dk = Ak −E(Ak |Fk−1), where Ak =∞∑t=0E(Xt+k |Fk)e

ıtλ, (3.6)

and Mn = ∑nt=1 Dt ∑t−1

j=1 αj−t Dj , where · denotes complex conjugate. Then

‖Ln −ELn − Mn‖m

32 n1/2‖X0‖2

4

≤ CV 1/2m (β), where Vm(β)

= max1−n≤i≤−1

β2i +m

−n−1

∑j=−1

|βj −βj−1|2.

Proof. Note that Ak = ∑mt=0E(Xt+k |Fk)eıtλ because E(Xt+k |Fk) = 0 if

t > m. Also Dk are m-dependent, and they form martingale differences with re-spect to Fk . Let � = eıλ, Uj = � j−t

E(Aj |Fj−1), c4 = ‖X0‖4, and βj = 0 if j ≥ 0or j ≤ −n. Observe that Xk = Ak −E(Ak+1|Fk)� and ‖A1‖4 ≤ 2mc4. Then∥∥∥∥∥t−8m

∑j=1

αj−t (X j − Dj )

∥∥∥∥∥=∥∥∥∥∥t−8m

∑j=1

βj−t (Uj −Uj+1)

∥∥∥∥∥


≤ Cmc4 maxj

|βj |+∥∥∥∥∥t−8m

∑j=1

(βj−t −βj−1−t )Uj

∥∥∥∥∥ . (3.7)

Note that Uj = ∑ml=1Pj−lUj and Pj−lUj , j ∈ Z, are martingale differences,∥∥∥∥∥t−8m

∑j=1

(βj−t −βj−1−t )Pj−lUj

∥∥∥∥∥2

=t−8m

∑j=1

(βj−t −βj−1−t )2‖P−lU0‖2. (3.8)

Because ∑ml=1 ‖P−lU0‖ ≤ m1/2(∑m

l=1 ‖P−lU0‖2)1/2 ≤ Cm3/2c4, by (3.7),∥∥∥∥∥t−8m

∑j=1

αj−t (X j − Dj )

∥∥∥∥∥≤ CV 1/2m (β)mc4. (3.9)

Let Qj = � j−tE( Aj |Fj−1). Using Xk = Ak − E( Ak+1|Fk)�, similarly, we

have∥∥∥∥∥ n

∑t= j+8m

αj−t (Xt − Dt )

∥∥∥∥∥=∥∥∥∥∥ n

∑t= j+8m

βj−t (Qj − Qj+1)

∥∥∥∥∥≤ CV 1/2m (β)mc4. (3.10)

Let W1,t = Xt ∑t−8mj=1 βj−t�

j−t (X j − Dj ). Then W1,t ,W1,t+4m,W1,t+8m, . . . are

martingale differences. By (3.9), ‖W1,t‖ ≤ CV 1/2m (β)mc2

4. Write � = V 1/2m

(β)m3/2n1/2c24. So∥∥∥∥∥ n

∑t=1

W1,t

∥∥∥∥∥≤4m−1

∑i=1

∥∥∥∥∥(n−i)/(4m)∑l=0

W1,i+4ml

∥∥∥∥∥≤ C�. (3.11)

Let Wt = W1,t + W2,t , where W2,t = Xt ∑t−1j=t−8m+1 βj−t�

j−t (X j − Dj ) are 12m-

dependent. As in (3.9), ‖W2,t‖ ≤ CV 1/2m (β)mc2

4. Similarly as (3.11), we have‖∑n

t=1(W2,t −EW2,t )‖ ≤ C�. By (3.11), ‖∑nt=1(Wt −EWt )‖ ≤ C�. Similarly,

using (3.10), we have‖∑nt=1(W

◦t −EW ◦

t )‖≤ C� for W ◦t = (Xt − Dt)∑t−1

j=1 αj−t Dj .Hence Proposition 2 follows. n

4. INEQUALITIES FOR m-DEPENDENT PROCESSES

As argued in Section 3, quadratic forms of (Xk) can be approximated by those ofm-dependent random variables. So probability inequalities under m-dependenceare useful for the asymptotic spectral estimation problem. Lemma 2, which fol-lows, is an easy consequence of Corollary 1.6 in Nagaev (1979) via a simpleblocking argument. We omit its proof. Proposition 3 is a Fuk–Nagaev-type in-equality for quadratic forms of m-dependent random variables. It is useful forproving the maximum deviation results in Section 2.3.


LEMMA 2. Let (Xk) be m-dependent with EXk = 0 and Xk ∈ Lp, p ≥ 2. LetWn = ∑n

i=1 Xi . Then for any Q > 0, there exists C1,C2 > 0 only depending on Qsuch that

P(|Wn| ≥ x) ≤ C1

( m

x2EW 2n

)Q

+C1 min

[m p−1

x p

n

∑k=1

‖Xk‖pp ,

n

∑k=1P

(|Xk | ≥ C2

x

m

)].

PROPOSITION 3. Let (Xt ) be m-dependent with EXt = 0, |Xt | ≤ M a.s.,m ≤ n, and M ≥ 1. Let Sk,l = ∑l+k

t=l+1 Xt ∑t−1s=1 an,t−s Xs , where l ≥ 0, l + k ≤ n

and assume max1≤t≤n |an,t | ≤ K0, max1≤t≤nEX2t ≤ K0, max1≤t≤nEX4

t ≤ K0 forsome K0 > 0. Then for any x ≥ 1, y ≥ 1, and Q > 0,

P(|Sk,l −ESk,l | ≥ x) ≤ 2e−y/4 +C1n3 M2

(x−2 y2m3(M2 + k)

n

∑s=1

a2n,s

)Q

+ C1n4 M2P

(|X0| ≥ C2x

ym2(M + k12 )

),

where C1,C2 > 0 are constants depending only on Q and K0.

Proof. Without loss of generality, we assume l = 0. Let Sk,0 = ∑kt=1 Xt ∑t−2m

s=1an,t−s Xs . Split the interval [1,k] into consecutive blocks H1, . . . , Hkn with samesize m. Here, for convenience, we assume kn = k/m ∈N. Let Sj = ∑t∈Hj

Xt ∑t−2ms=1

an,t−s Xs , 1 ≤ j ≤ kn . Then {Sj ; j = 1,3, . . .} and {Sj ; j = 2,4, . . .} are two setsof martingale differences. Let Gj = σ(Si ; 1 ≤ i ≤ j). By Freedman’s inequality(see Freedman, 1975), we have

P(|Sk,0| ≥ 2x) ≤ 2e−y/4 +kn

∑j=1P(|Sj | ≥ x/y)

+ P

(∣∣∣∣∣ kn

∑j=1E[Sj I{Sj ≥ x/y}|Gj−2

]∣∣∣∣∣≥ x

)

+ P

(kn

∑j=1E[S 2

j |Gj−2] ≥ x2/y

)=: 2e−y/4 + In + IIn + IIIn .

Because |Sj | ≤ CnmM2, we have IIn ≤ Cx−1nmM 2 ∑knj=1P(|Sj | ≥ x/y). By

Lemma 2,

P

(|Sj | ≥ x/y

)≤ m max

1≤t≤nP

(∣∣∣∣∣ t

∑s=1

an,t−s Xs

∣∣∣∣∣≥ x

ymM

)


≤ C1m

(x−2 y2m3 M2

n

∑s=1

a2n,s

)Q

+C1mnP

(|X0| ≥ C2x

ym2 M

).

Because E(S 2j |Gj−2) = ∑t1,t2∈Hj

E(Xt1 Xt2)∑t1−2ms=1 an,t−s Xs ∑t2−2m

s=1 an,t−s Xs , wehave

IIIn ≤kn

∑j=1P

(E[S2

j |Gj−2] ≥ mx2/(yk))

≤kn

∑j=1

∑t1,t2∈Hj

P

(∣∣∣∣ t1−2m

∑s=1

an,t1−s Xs

t2−2m

∑s=1

an,t2−s Xs

∣∣∣∣≥ x2

ymk

)

≤ 2km max1≤t≤n

P

(∣∣∣∣ t

∑s=1

an,t−s Xs

∣∣∣∣≥ x

(ymk)12

)

≤ C1km

(x−2ym2k

n

∑s=1

a2n,s

)Q

+C1kmnP

(|X0| ≥ C2x

(ym3k)12

).

It remains to consider S�k,0 := ∑k

t=1 Xt ∑t−1s=t−2m+1 an,t−s Xs . By Lemma 2, we

have

P(|S�

k,0 −ES�k,0| ≥ x

)≤ C1

(x−2m2

n

∑s=1

a2t

)Q

+C1kmP

(|X0| ≥ C2x

m2 M

).

The proof is now complete. n

5. A GENERAL CLT FOR QUADRATIC FORMS

In this section we establish a very general CLT for quadratic forms of stationaryprocesses, which can be used for proving Theorem 2. For quadratic forms of in-dependent random variables see de Jong (1987), Mikosch (1991), ten Vregelaar(1991), and Gotze and Tikhomirov (1999), among others. For more referencessee Wu and Shao (2007), which gives a CLT for quadratic forms of martingaledifferences. Theorem 6 imposes the very mild dependence condition 0,4 < ∞,and it allows a wide class of weights an, j . Recall that ω(u) = 2 if u/π ∈ Z andω(u) = 1 if u/π �∈ Z.

THEOREM 6. Let an, j = bn, j eı jλ, where λ ∈ R, bn, j ∈ R with bn, j = bn,− j ,and

Tn = ∑1≤ j, j ′≤n

an, j− j ′ X j X j ′ and σ 2n = ω(λ)

n

∑k=1

n

∑t=1

b2n,t−k . (5.1)


Assume that EX0 = 0, E|X0|4 < ∞, 0,4 < ∞, and

max0≤t≤n

b2n,t = o(ς2

n ), where ς2n =

n

∑t=1

b2n,t ; (5.2)

nς2n = O(σ 2

n ); (5.3)

n

∑k=1

k−1

∑t=1

∣∣∣∣∣ n

∑j=1+k

an,k− j an,t− j

∣∣∣∣∣2

= o(σ 4n ); (5.4)

n

∑k=1

|bn,k −bn,k−1|2 = o(ς2n ). (5.5)

Then for 0 ≤ λ < 2π , σ−1n (Tn −ETn) ⇒ N (0,4π2 f 2(λ)).

Proof. We shall apply Propositions 1 and 2 with αj = an, j = bn, j eı jλ. RecallPropositions 1 and 2 for Ln, Ln, Dk , and Mn . Let Ln and Mn be the complexconjugates of Ln and Mn . Note that Tn = Ln + Ln +an,0 ∑n

j=1 X2j . BecauseE|X0|4

< ∞ and 0,4 < ∞, we have ‖∑nj=1 X2

j − nγ0‖ ≤ C√

n‖X0‖40,4 = O(√

n).Therefore, because dm,2p → 0 as m → ∞, by (5.2), (5.3), and Propositions 1and 2, it suffices to show that

Mn + Mn

σn⇒ N (0,4π2 f 2(λ)), where 2π f (λ) =

m

∑j=−m

γj cos( jλ), (5.6)

holds for every m. Here γj = E(X0 X j ). Let U�t = ∑t−1

j=(t−4m+1)∨1 an, j−t Dj . By

(5.2) and (5.3), we have ‖∑nt=1 DtU�

t ‖ = O(√

n)max1≤t≤n |bn,t |. So it remainsto show that

1

σn

n

∑t=1+4m

(DtUt + DtUt ) ⇒ N (0,4π2 f 2(λ)), where Ut =t−4m

∑j=1

an, j−t Dj .

(5.7)

Because ∑nt=1+4m ‖DtUt‖4

4 = O(n)ς4n = o(σ 4

n ), the Lindeberg condition easilyfollows. By the martingale CLT (see Hall and Heyde, 1980), (5.7) holds if

1

σ 2n

n

∑t=1+4m

E[(DtUt + DtUt )2|Ft−1] → 4π2 f 2(λ) in probability (5.8)

For the rest of the proof, we shall verify (5.8). Let −m ≤ l ≤ m −1. Then∥∥∥∥∥ n

∑t=1+4m

Pt+l(DtUt + DtUt )2

∥∥∥∥∥2

=n

∑t=1+4m

‖Pt+l(DtUt + DtUt )2‖2

≤ 4n

∑t=1+4m

‖Dt‖44‖Ut‖4

4.


Hence ‖∑nt=1+4mPt+l(DtUt + DtUt )

2‖ = o(σ 2n ). Because Dt is Ft−m,t measur-

able and Ut is Ft−3m measurable, E(D2t U 2

t |Ft−m−1) = U 2t E(D2

t ), (5.8) is thenreduced to

1

σ 2n

n

∑t=1+4m

[U 2t E(D2

t )+ U 2t E(D2

t )+2|Ut |2E(|D2t |)] → ‖D0‖4 in probability

(5.9)

by noting that ‖D0‖2 = ∑mj=−m γj eı jλ = 2π f (λ). For the rest of the proof we

shall only deal with the case of λ �= 0,π because the case of λ = 0,π can besimilarly proved. Using the argument of Lemma 3 in Wu and Shao (2007), under(5.2)–(5.4), we can prove that ‖∑n

t=1+4m(U 2t −EU 2

t )‖ = o(σ 2n ). So (5.9) is further

reduced to

1

σ 2n

n

∑t=1+4m

[E(U 2t )E(D2

t )+E(U 2t )E(D2

t )+2E(|Ut |2)E(|D2t |)] → ‖D0‖4. (5.10)

Clearly E(U 2t ) = ∑t−4m

j=1 a2n, j−tED2

j . By summation by parts and (5.5), because

|∑ jl=1 e2ılλ| ≤ 1/|sinλ|, we have maxt≤n |∑t

j=1 a2n, j−t | = o(ς2

n ) and ∑nt=1+4m

|E(U 2t )| = o(σ 2

n ). So (5.10) follows from (5.2) and (5.3) because E(|Ut |2) =∑t−4m

j=1 b2n, j−tE(|D2

t |). n

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TECHNICAL APPENDIX AND PROOFS

A.1. Proofs of the Results in Sections 2.1–2.3.

Proof of Theorem 1. Recall (3.1) for Xt = Xt,m , which are m-dependent. Let an,t =K (t/Bn)cos(tθ). For 4m +1 ≤ t ≤ n let Yt = Xt ∑t−4m

s=1 an,t−s Xs and Rm = ∑nt=1+4m Yt .

By Condition 1, ∑s∈Z a2n,s = O(Bn). By independence and Lemma 1,

‖Yt‖p = ‖Xt‖p

∥∥∥∥∥t−4m

∑s=1

an,t−s Xs

∥∥∥∥∥p

≤ ‖X0‖pCp0,p

(t−4m

∑s=1

a2n,t−s

)1/2

= O(B1/2n ). (A.1)


Let Jl = (n − l)/(4m). Because Yt ,Yt+4m ,Yt+8m , . . ., are Lp martingale differences,

‖Rn‖p ≤4m

∑l=1

∥∥∥∥∥ Jl

∑j=1

Yl+4mj

∥∥∥∥∥p

=4m

∑l=1

J 1/2l O(B1/2

n ) = O[(mnBn)1/2). (A.2)

Let γk = E(X0 Xk), gn(θ) = 2π fn(θ) and

gn(θ) = 1

n

n

∑t=1

X2t + 2

n

n

∑t=2

Xt

t−1

∑s=1

an,t−s Xs

= 1

n

n

∑t=1

X2t + 2

n

n

∑t=2

Xt

t−1

∑s=max(1,t−4m+1)

an,t−s Xs + 2Rn

n. (A.3)

By the ergodic theorem, for 1 ≤ l ≤ 4m, ‖n−1 ∑nt=1 Xt Xt+l − γl‖p/2 → 0. So

limn→∞‖gn(θ)−2n−1 Rn −E[gn(θ)−2n−1 Rn]‖p/2 = 0. (A.4)

Let In(u) = (2πn)−1|Sn(u)|2, where Sn(u) = ∑nk=1 Xkeıku . By Lemma 1, ‖Sn(u) −

Sn(u)‖p = O(n1/2)m,p , ‖Sn(u)‖p +‖Sn(u)‖p = O(n1/2). Hence ‖In(u)− In(u)‖p/2 =O(1)m,p . Because fn(θ) = ∫R K (u) In(B−1

n u + θ)du, p/2 ≥ 1, and∫R |K (u)|du < ∞,

‖ fn(θ)− fn(θ)‖p/2 ≤∫R

|K (u)|‖In(B−1n u + θ)− In

(B−1

n u + θ)

‖p/2du = O(1)m,p.

So |E[ fn(θ)− fn(θ)]| = O(m,p). By (A.2) and (A.4), because Bn = o(n), limn→∞ ‖ fn(θ)−E[ fn(θ)]‖p/2 = 0. Hence limn→∞ ‖ fn(θ)−E fn(θ)‖p/2 = 0 in view of

‖ fn(θ) − E fn(θ)‖p/2 ≤ ‖ fn(θ)− fn(θ)‖p/2 +|E[ fn(θ)− fn(θ)]|+‖ fn(θ)

− E[ fn(θ)]‖p/2

and m,p → 0 as m → ∞. It is well known in time series analysis that, under (1.1),Bn → ∞, and limu→0 K (u) = 1, the bias E fn(θ)− f (θ) → 0. So (2.4) follows. n

Proof of Theorem 2. We shall apply Theorem 6 to bn, j = K ( j/Bn) and an, j = bn, j

eı jλ. By Condition 2, easy calculations show that 2∑nk=1 K 2(k/Bn) ∼ Bnκ and ∑n

k=1∑n

t=1 b2n,t−k ∼ nBnκ . So (5.2) and (5.3) hold. Let M be a positive integer that will be

specialized later. Because Bn = o(n), by Schwarz’s inequality,

n

∑k=1

k−1

∑t=1∨(k−M Bn)

(n

∑j=1+k

bn,k− j bn,t− j

)2

= O(nB3n ) = o(σ 4

n ). (A.5)

Using Schwarz’s inequality again, we can get

n

∑k=1

k−M Bn

∑t=1

(n

∑j=1+k

bn,k− j bn,t− j

)2

≤ Cn2 Bn

n

∑k=M Bn

K 2(k/Bn) ≤ τM n2 B2n (A.6)


with τM → 0 as M → ∞. Combining (A.5) and (A.6), we have (5.4). It remains to prove(5.5). Because K is continuous at all but a finite number of points, it can be easily ob-tained that |K (k/Bn)− K ((k − 1)/Bn)| = o(1) uniformly for |k| ≤ M Bn , except for εBn

points k ∈ [−M Bn, M Bn], where ε > 0 is an arbitrary number. Thus, ∑M Bnk=1 {K (k/Bn)−

K [(k −1)/Bn]}2 = o(Bn). Moreover, it is easy to see that ∑nk=M Bn

{K (k/Bn)− K [(k −1)/

Bn]}2 ≤ τM Bn . Hence (5.5) is proved. n

Remark A.1. It is easily seen that Theorem 2 also holds if the requirement that K iscontinuous at all but a finite number of points in Condition 2 is replaced by the followingone: K has bounded variation. The two conditions have different ranges of applicability.

A.2. Proofs of Theorems 3–5. Let an,t = K (t/Bn)cos(tλ). Recall (5.1) for Tn and de-fine Tn,m by replacing {Xt } in (5.1) by {Xt,m}. Let gn(λ) = Tn −ETn −∑n

k=1(X2k −EX2

k )

and gn,m(λ) = Tn,m −ETn,m − ∑nk=1(X2

k,m −EX2k,m). Then 2πn{ fn(λ) −E fn(λ)} =

gn(λ)+ ∑nk=1(X2

k −EX2k ). The proofs of Theorems 3–5 are quite complicated, and they

are based on a series of lemmas. Let τn = √nBn/log Bn .

LEMMA A.1. Suppose that the conditions of Theorem 3 hold. Then for any 0 < C < 1there exists γ ∈ (0,C) such that, for m = nγ ,

max0≤i≤Bn

|gn(λ∗i )− gn,m(λ∗

i )| = oP(√

nBn/ log Bn

).

Remark A.2. By Proposition 1, Lemma A.1 also holds under conditions of Theorem 5.

Proof. Let � ∈ (0,1) be fixed and be sufficiently close to 1. Let sl = n�l , 1 ≤ l ≤ r ,and r ∈ N be such that 0 < �r < C. Let r0(n) ∈ N satisfy 1 ≤ r0(n) ≤ r and sr0(n) < Bn ≤sr0(n)−1. By Markov’s inequality and Proposition 1 and because p > 4,

P

(max

0≤i≤Bn|gn(λ∗

i )− gn,s1(λ∗i )| ≥ τn

)≤ (1+ Bn) max

0≤i≤Bn

E|gn(λ∗i )− gn,s1(λ

∗i )|p/2

(nBn)p/4(log Bn)−p/2

≤ C Bnd p/2s1,p(log Bn)p/2

≤ Cnδ−T1�p/2(logn)p/4−2 = o(1).

So we only need to show that, for every 1 ≤ l ≤ r − 1, max0≤i≤Bn |gn,sl (λ∗i ) − gn,sl+1

(λ∗i )| = oP

(√nBn/ log Bn

). Let Yt,m(λ) = Xt,m ∑t−1

s=1 Xs,man,t−s and note that, for any1 ≤ l ≤ r , Yt,sl (λ), 1 ≤ t ≤ n, are (Bn + sl )-dependent. Split the interval [1,n] into con-secutive blocks H1, H2, . . ., Htn with equal length Bn + sl , where the number of intervalstn ∼ n/(Bn + sl ) and the last interval may be incomplete. For convenience we assume thatthe length of the last interval is also Bn + sl . Define

u j (λ) = ∑t∈Hj

(Yt,sl (λ)−Yt,sl+1(λ)), uj (λ) = u j (λ)−Eu j (λ), 1 ≤ j ≤ tn . (A.7)

Then u1(λ), . . . ,utn (λ) are 1-dependent and gn,sl (λ) − gn,sl+1(λ) = 2∑tnj=1 uj (λ). By

Lemma 2, for any large Q and 1 ≤ l ≤ r −1,

P

(max

0≤i≤Bn|gn,sl (λ

∗i )− gn,sl+1(λ

∗i )| ≥ τn

)


≤ CBn

∑i=0

(∑tn

j=1E|uj (λ∗i )|2

τ2n

)Q

+CBn

∑i=0

tn

∑j=1

P(|uj (λ

∗i )| ≥ CQτn

). (A.8)

Because dn,p = O(n−T1), by Proposition 1, maxλ∈Rmax1≤ j≤tn E|uj (λ)|2 = O(

Bn(Bn+sl )s

−2T1l+1

). Let Q ∈N be sufficiently large. Then the first term in the preceding expression

is o(1). It remains to show that the second one is also o(1). We first deal with the case

1 ≤ l ≤ r0(n)− 1. Because sl ∼ sl+1nρl (1−ρ) and sr0(n)−1 ≥ Bn , we have tn � n/sl . ByProposition 1 and Markov’s inequality,

Bn

∑i=0

tn

∑j=1

P(|uj (λ

∗i )| ≥ CQτn

)≤ C Bnns−1l s p/4

l s−pT1/2l+1 n−p/4(log Bn)p/2

≤ C Bns p/4−1−pT1/2l n1−p/4+ε� (log Bn)p/2 =: F1, (A.9)

where ε� → 0 as � → 1. If 4 < p ≤ 4+4δ, then p/4−1− pT1/2 ≤ 0, and hence

F1 ≤ C B p/4−pT1/2n n1−p/4+ε� (log Bn)p/2 ≤Cn0∨{δ(p/4−pT1/2)}+1−p/4+ε� (log Bn)p/2

= o(1),

where we have used sl ≥ Bn and Condition 5(a). If p > 4+4δ and p/4−1− pT1/2 ≤ 0,then F1 = o(1). Finally, if p > 4 + 4δ and p/4 − 1 − pT1/2 > 0, then because sl ≤ n, wehave F1 ≤ Cnδ−pT1/2+ε� (log Bn)p/2 = o(1). Hence F1 = o(1) when 1 ≤ l ≤ r0(n)− 1.We now deal with the case r0 ≤ l ≤ r −1. For r0(n) ≤ j ≤ r , r0(n) ≤ l ≤ r −1, let

Ut, j =t−sl−1

∑s=1

an,t−s Xs,sj , Yt,sl = Xt,sl Ut,l − Xt,sl+1Ut,l+1.

Let j1 = min{k : k ∈ Hj } and j2 = max{k : k ∈ Hj }. Because Xt,i = ∑tk=t−sl

Pk Xt,i fori = sl ,sl+1, we have

u′j (λ) : = ∑

t∈Hj

Yt,sl =j2

∑k= j1−sl

W k,l , where W k,l

=(k+sl )∧ j2

∑t=k∨ j1

(Pk Xt,sl Ut,l −Pk Xt,sl+1Ut,l+1).

Note that W k,l , j1 − sl ≤ k ≤ j2, are martingale differences; by Lemma 1,

maxλ∈R ‖W k,l‖p ≤ max

λ∈R(k+sl )∧ j2

∑t=k∨ j1

‖Pk Xt,sl ‖p‖Ut,l −Ut,l+1‖p

+maxλ∈R

(k+sl )∧ j2

∑t=k∨ j1

‖Pk(Xt,sl − Xt,sl+1)‖p‖Ut,l+1‖p = O(B1/2n ).


By Markov’s inequality, because p > 2/(1− δ), we have

Bn max0≤i≤Bn

tn

∑j=1

P

(|u′

j (λ∗i )| ≥ τn

)≤ Cn1−p/2 B−1

n (log Bn)2pj2

∑k= j1−sl

maxλ∈RE|W k,l |p

≤ Cn1−p/2 B p/2n (log Bn)2p = o(1). (A.10)

Putting Yt,sl = Yt,sl (λ) − Yt,sl+1(λ) − Yt,sl , then uj (λ) = u′j (λ) + ∑t∈Hj (Yt,sl −EYt,sl ).

By Lemma 1, we have E|Yt,sl |p/2 ≤ Cs p/4l s−T2 p/2

l+1 . Because Yt,sl , 1 ≤ t ≤ n, are sl -dependent, by Lemma 2, for Q large enough,

Bn max0≤i≤Bn

tn

∑j=1

P

(|uj (λ

∗i )−u′

j (λ∗i )| ≥ τn

)

≤ C Bn max0≤i≤Bn

tn

∑j=1

(sl ∑t∈Hj E|Yt,sl |2nBn(log Bn)−2

)Q

+ C Bn(nBn)−p/4(log Bn)p/2s p/2−1l max

0≤i≤Bn

tn

∑j=1

∑t∈Hj

E|Yt,sl |p/2

≤ Cs3p/4−1l s−pT2/2

l+1 (nBn)−p/4+1(log Bn)p/2 = o(1), (A.11)

where the last relation is due to sl < Bn , sl ∼ sl+1nρl (1−ρ) and Condition 5(b). By(A.8)–(A.11), Lemma A.1 follows. n

LEMMA A.1*. Under the conditions of Theorem 4, the conclusion in Lemma A.1 holds.

Proof. By the argument in the proof of Lemma A.1, we only need to show that, for ev-

ery r0(n) ≤ l ≤ r −1, max0≤i≤Bn |gn,sl (λ∗i )− gn,sl+1(λ

∗i )| = oP

(√nBn/ log Bn

). Recall

(A.7) for uj (λ). Let u′j (λ) = uj (λ)I

{|uj (λ)| ≤ √

nBn/(log Bn)3}

. Then

P

(max

0≤i≤Bn

∣∣∣∣∣ tn

∑j=1

uj (λ∗i )

∣∣∣∣∣≥ τn

)≤ P(Gn ≥ τn)+

tn

∑j=1

P

(Pj ≥

√nBn

(log Bn)3

),

where Gn = max0≤i≤Bn |∑tnj=1 u′

j (λ∗i )| and Pj = max0≤i≤Bn |uj (λ

∗i )|. By Proposition 1,

∑tnj=1E|u′

j (λ)|2 ≤ CnBnd2sl ,4

, and

tn

∑j=1

|Eu′j (λ)| ≤

tn

∑j=1

E|uj (λ)|2(nBn)−1/2(log Bn)3

≤ C(nBn)1/2d2sl ,4(log Bn)3 = o

(√nBn/ log Bn

),

we have, by Bernstein’s inequality, P(Gn ≥ τn) ≤ C Bn exp(−C(log Bn)2) = o(1). To fin-ish the proof of Lemma A.1∗, we only need to show that

�n :=tn

∑j=1

P

(Pj ≥√nBn/(log Bn)3

)= o(1).


Recall Yt,m(λ) = Xt,m ∑t−1s=1 Xs,man,t−s . Because for 2 ≤ j ≤ tn , uj (λ)

D= u2(λ) and

u2(λ) =2Bn+2sl

∑t=1

(Yt,sl (λ)−Yt,sl+1(λ))−Bn+sl

∑t=1

(Yt,sl (λ)−Yt,sl+1(λ)).

Let Sn,1(u) = ∑2Bn+2slk=1 Xk,sl e

ıku and Sn,2(u) = ∑2Bn+2slk=1 Xk,sl+1 eıku , and similarly de-

fine S′n,1(u), S′

n,2(u), by replacing 2Bn +2sl in Sn,1(u) and Sn,2(u) by Bn + sl . Then

2u2(λ) = Bn

∫ ∞−∞

K (Bn(u −λ))[|Sn,1(u)|2 −|Sn,2(u)|2

]du

− Bn

∫ ∞−∞

K (Bn(u −λ))[|S′

n,1(u)|2 −|S′n,2(u)|2

]du

−2Bn+2sl

∑t=Bn+sl+1

X2t,sl

+2Bn+2sl

∑t=Bn+sl+1

X2t,sl+1

.

Let Qn,k(u) = |Sn,k(u)|2 −E|Sn,k(u)|2, Q′n,k(u) = |S′

n,k(u)|2 −E|S′n,k(u)|2, k = 1,2,

Wn = maxu∈R |Qn,1(u)− Qn,2(u)|, and W ′n = maxu∈R |Q′

n,1(u)− Q′n,2(u)|. Then

2maxλ∈R |u2(λ)| ≤ (Wn + W ′

n)

∫ ∞−∞

|K (u)|du

+∣∣∣∣∣ 2Bn+2sl

∑t=Bn+sl+1

(X2t,sl

−EX2t,sl

)

∣∣∣∣∣+∣∣∣∣∣ 2Bn+2sl

∑t=Bn+sl+1

(X2t,sl+1

−EX2t,sl+1

)

∣∣∣∣∣ .Define μn = √

nBn/(log Bn)3 and νn = (nBn)1/4/(log Bn)3. Then

�n ≤tn

∑j=1

P(Wn ≥ 2−1μn)+tn

∑j=1

P(W ′n ≥ 2−1μn)+o(1)

≤tn

∑j=1

P

(max

0≤u≤2π(|Sn,1(u)|+ |Sn,2(u)|) ≥ νn

)

+tn

∑j=1

P

(max

0≤u≤2π(|S′

n,1(u)|+ |S′n,2(u)|) ≥ νn

)+o(1).

Let li = i/n2, 0 ≤ i ≤ 2πn2. Then

max0≤u≤2π

|Sn,1(u)| ≤ max0≤i≤2πn2

|Sn,1(li )|+Cn−2 Bn

Bn+sl

∑k=1

|Xk,sl |.

So it suffices to show that ∑tnj=1P(max0≤i≤2πn2 |Sn,1(li )| ≥ νn) = o(1). Note that

Sn,1(u) =2Bn+2sl

∑j=1−sl

(2Bn+2sl )∧( j+sl )

∑k=1∨ j

Pj (Xk,sl )eıku =:

2Bn+2sl

∑j=1−sl

Rn, j (u).


Set Rn, j (u) = Rn, j (u)I{|Rn, j (u)| ≤ (nBn)1/4/(log Bn)5

}. Then

tn

∑j=1

P

(max

0≤u≤2π

∣∣∣∣∣2Bn+2sl

∑j=1−sl

Rn, j (u)− Rn, j (u)

∣∣∣∣∣≥ νn

)

≤ Cn(log Bn)18

Bn(nBn)

2Bn+2sl

∑j=1−sl

∥∥∥∥∥(Bn+sl )∧( j+sl )

∑k=1∨ j

∣∣Pj (Xk,sl )∣∣∥∥∥∥∥

4

4

≤ C(log Bn)18

Bn= o(1),

where Rn, j (u) = Rn, j (u)−E(Rn, j (u)|Fj−1). Also,∥∥∥∥∥maxu∈R

2Bn+2sl

∑j=1−sl

E(|Rn, j (u)|2|Fj−1)

∥∥∥∥∥p/2

≤∥∥∥∥∥max

u∈R2Bn+2sl

∑j=1−sl

E(|Rn, j (u)|2|Fj−1)

∥∥∥∥∥p/2

≤2Bn+2sl

∑j=1−sl

((2Bn+2sl )∧( j+sl )

∑k=1∨ j

δk− j,p

)2

≤ C Bn .

We can get, by Freedman’s inequality and because p > 4,

tn

∑j=1

P

(max

0≤i≤2πn2

∣∣∣∣∣ Bn+sl

∑j=1−sl

Rn, j (li )

∣∣∣∣∣≥ νn

)

≤ Cn3 B−1n exp(−C(log Bn)2)+nB−1

n B p/2n (nBn)−p/4(log Bn)5p/2 = o(1).

The proof is now complete. n

For Lemmas A.2–A.4, we need to introduce truncation. Let α < 14 be close to 1

4 suffi-ciently and m = nγ , where γ is small enough. Define

X ′t,m = Xt,m I{|Xt,m | ≤ (nBn)α}, Xt,m = X ′

t,m −EX ′t,m ,

gn,m(λ) = 2n

∑t=2

Xt,m

t−1

∑s=1

an,t−s Xs,m −2En

∑t=2

Xt,m

t−1

∑s=1

an,t−s Xs,m .

Let pn = B1+βn ,qn = Bn +m, and kn = n/(pn +qn), where β > 0 is sufficiently close

to zero. Split the interval [1,n] into alternating big and small blocks Hj and Ij by

Hj = [( j −1)(pn +qn)+1, j pn + ( j −1)qn];Ij = [ j pn + ( j −1)qn +1, j (pn +qn)];1 ≤ j ≤ kn, Ikn+1 = [kn(pn +qn)+1,n].

Set Y t,m(λ) = Xt,m ∑t−1s=1 an,t−s Xs,m . For 1 ≤ j ≤ kn +1 let

uj (λ) = ∑t∈Hj

(Y t,m(λ)−EY t,m(λ) and v j (λ) = ∑t∈Ij

(Y t,m(λ)−EY t,m(λ)). (A.12)


Because K (·) is bounded, we have

Emaxλ∈R∣∣∣gn,m(λ)− gn,m(λ)

∣∣∣≤ CEn

∑t=2

|Xt,m |t−1

∑s=1∨(t−Bn)

|Xs,m − Xs,m | (A.13)

+ CEn

∑t=2

|Xt,m − Xt,m |t−1

∑s=1∨(t−Bn)

|Xs,m |.

Recall τn = √nBn/log Bn . By independence and because X0 ∈ Lp, p > 4,

E

(n

∑t=m+1

|Xt,m |t−1

∑s=1∨(t−Bn)

|Xs,m − Xs,m |)

(A.14)

≤ E(

n

∑t=m+1

|Xt,m |t−m

∑s=1∨(t−Bn)

|Xs,m − Xs,m |)

+E(

n

∑t=2

|Xt,m |t−1

∑s=(t−m+1)∨1

|Xs,m − Xs,m |)

≤ C(nBn)1−(p−1)α +Cnm(nBn)−(p−2)α = o(τn).

Similar arguments yield to

E

(n

∑t=2

|Xt,m − Xt,m |t−1

∑s=1∨(t−Bn)

|Xs,m |)

= o(τn). (A.15)

Combining (A.13)–(A.15), we get

Emaxλ∈R∣∣∣gn,m(λ)− gn,m(λ)

∣∣∣= o(τn). (A.16)

LEMMA A.2. Assume EX0 = 0 and EX40 < ∞. Recall τn = √

nBn/ log Bn. We have

max|i |≤Bn|�i | = OP (τn) , where �i =

kn+1

∑j=1

v j (λ∗i ).

Proof. Because v j (λ∗i ), 1 ≤ j ≤ kn +1, are independent, by Lemma 2, for all large Q,

P(|�i | ≥ τn) ≤ C

⎛⎝∑kn+1j=1 Ev2

j (λ∗i )

nBn(log Bn)−2

⎞⎠Q

+Ckn+1

∑j=1

P(|v j (λ

∗i )| ≥ CQτn

).

In Proposition 3 we let x = CQτn , M = (nBn)α , k = Bn +m, m = nγ , and y = (log Bn)2.Then for all c > 0, P(|v j (λ

∗i )| ≥ CQτn) = O(n−c). Because P(max|i |≤Bn |�i | ≤ τn) ≤

∑|i |≤Bn P(|�i | ≥ τn), by elementary manipulations, the lemma follows. n


LEMMA A.3. Assume that EX0 = 0 and EX40 < ∞. For 1 ≤ j ≤ kn let

u j (λ) = uj (λ)I

{∣∣uj (λ)∣∣≤ √

nBn

(log Bn)4

}−Euj (λ)I

{∣∣uj (λ)∣∣≤ √

nBn

(log Bn)4

}(A.17)

be the truncated version of uj (λ). Then

P

(max

0≤i≤Bn

∣∣∣∣∣ kn

∑j=1

(uj (λ∗i )− u j (λ

∗i ))

∣∣∣∣∣≥√nBn(log Bn)−1

)= o(1).

Proof. As in the proof of Lemma A.2, we can get

P

(|uj (λ

∗i )| ≥ CQ

√nBn

(log Bn)4

)= O(n−c) (A.18)

for any large c > 0. The lemma immediately follows. n

LEMMA A.4. Assume that EX0 = 0 and EX40 < ∞. We have for any x > 0,

P

(maxi∈B

∣∣∣∣∣ kn

∑j=1

u j (λ∗i )

∣∣∣∣∣≥ x√

nBn log Bn

)= o(1),

where B= {|i | ≤ (log Bn)2}∪{Bn − (log Bn)2 ≤ |i | ≤ Bn}.Proof. The lemma easily follows from Bernstein’s inequality. n

LEMMA A.5. Suppose that EX0 = 0, EX40 < ∞, and dn,4 = O((logn)−2).

(i) We have

|E[gn(λ1)−Egn(λ1)][gn(λ2)−Egn(λ2)]| ≤ CnBn(log Bn)−2

uniformly on {(λ1,λ2) : 0 ≤ λk ≤ π − B−1n (log Bn)2, k = 1,2, and |λ1 − λ2| ≥

B−1n (log Bn)2}.

(ii) For αn > 0 with limsupn→∞ αn < 1, we have, uniformly on {(λ1,λ2) : B−1n (log

Bn)2 ≤ λk ≤ π − B−1n (log Bn)2, k = 1,2, and |λ1 −λ2| ≥ B−1

n }, that

|E[gn(λ1)−Egn(λ1)][gn(λ2)−Egn(λ2)]| ≤ 4π2αnnBn f (λ1) f (λ2)κ.

(iii) We have uniformly on {B−1n (log Bn)2 ≤ λ ≤ π − B−1

n (log Bn)2} that

|E[gn(λ)−Egn(λ)]2 −4π2nBn f 2(λ)κ| ≤ CnBn(log Bn)−2.

Proof.

(i) Let m = nε with ε > 0 being small enough. Then dm,4 = O((log Bn)−2). Let

an, j = K ( j/Bn)eı jλ. Denote Dk in (3.6) by Dk,λ. Let Mn(λ) = ∑nt=1 Dt,λ ∑t−1

j=1an, j−t Dj,λ. By Propositions 1 and 2, it suffices to verify that

rn,λ1,λ2 := |E[Mn(λ1)+ Mn(λ1)][Mn(λ2)+ Mn(λ2)]| ≤ CnBn

(log Bn)2 .


Let Nn(λ) = ∑nt=1 Dt,λ ∑t−m−1

j=1 an, j−t Dj,λ. Because Dt,λ, t ≥ 1 are martingaledifferences, elementary manipulations of trigonometric identities show that

rn,λ1,λ2 = 2|ED0,λ1 D0,λ2 |2n

∑t=1

t−m−1

∑s=1

K 2((t − s)/Bn)cos((t − s)(λ1 +λ2))

+2|ED0,λ1 D0,λ2 |2n

∑t=1

t−m−1

∑s=1

K 2((t − s)/Bn)

×cos((t − s)(λ1 −λ2)). (A.19)

Using the identity 1 + 2∑nk=1 cos(kλ) = sin((n + 1)λ/2)/sin(λ/2), by the sum-

mation by parts formula and Condition 3, it follows that∣∣∣∣∣ n

∑t=1

t−m−1

∑s=1

K 2((t − s)/Bn)cos((t − s)(λ1 ±λ2))

∣∣∣∣∣≤ Cnm+CB2

n +n

∑t=Bn+m+1

∣∣∣∣∣ Bn

∑s=1

K 2(s/Bn)cos s(λ1 ±λ2)

∣∣∣∣∣≤ Cnm+CB2

n +CnBn/(log Bn)2,

which by (A.19) implies rn,λ1,λ2 = O(nBn/(log Bn)2). By orthogonality, ‖Mn(λ) − Nn(λ)‖ = O(

√nm), we have |rn,λ1,λ2 | ≤ |rn,λ1,λ2 | + O(n

√Bnm + √

nm

Bn) = O(nBn/(log Bn)2), and hence (i) holds.

(ii) As in the proof of Lemma 3.2(ii) in Woodroofe and Van Ness (1967), usingCondition 3, we can show that

limsupn→∞

2(nBn)−1n

∑t=1

t−m−1

∑s=1

K 2((t − s)/Bn)cos((t − s)(λ1 −λ2)) < κ.

Hence (ii) follows.

(iii) Recall that ‖D0,λ‖2 = ∑mj=−m E(X0,m X j,m)eı jλ. From the proof of (i) and Con-

dition 3, we see that for B−1n (log Bn)2 ≤ λ ≤ π − B−1

n (log Bn)2,

rn(λ,λ) = O(nBn/(log Bn)2)+‖D0,λ‖42n

Bn

∑s=−Bn

K 2(s/Bn)

= O(nBn/(log Bn)2)+4π2 f 2(λ)nBnκ.

Hence (iii) holds. n

LEMMA A.6. Recall (A.17) for u j (λ) and set En = Bn − (log Bn)2. Under the condi-tions of Theorem 3 or 4 or 5, we have

P

(max(log Bn)2≤i≤En

|∑knj=1 u j (λ

∗i )|2

4π2nBn f 2(λ∗i )κ

−2log(Bn)+ log(π log Bn) ≤ x

)→ e−e−x/2

.


Proof. For convenience we assume κ = ∫ 1−1 K 2(t)dt = 1. LetVn = ∑knj=1 Vj , where

Vj = ( f −1(λ∗i1

)u j (λ∗i1

), . . . , f −1(λ∗id

)u j (λ∗id

)), 1 ≤ j ≤ kn,

(log Bn)2 ≤ i1 < · · · < id ≤ En .

By Fact 2.2 in Einmahl and Mason (1997), there exist independent centered normal randomvectors N1, . . . , Nkn with Cov(Vj ) = Cov(Nj ), 1 ≤ j ≤ kn , such that

P(|Vn −Nn | ≥ τn) = O(

e−(log Bn)3)

,

whereNn = ∑knj=1 Nj . For z = (z1, . . . , zd ) define |z|d = min{zi : 1 ≤ i ≤ d}. Then

P

(|Vn |d ≥ yn

√nBn

)≤ P(|Nn |d ≥ yn

√nBn − τn

)+ O

(e−(log Bn)3

). (A.20)

Let Id be a d ×d identity matrix. We claim that

|Cov(Nn)−4π2nBnId | = O(nBn/(log Bn)2). (A.21)

To prove (A.21), we first show that for |ik − il | ≥ (log Bn)2/Bn ,∣∣∣∣∣E kn

∑j=1

u j (λ∗ik

)kn

∑j=1

u j (λ∗il )

∣∣∣∣∣= O(nBn/(log Bn)2). (A.22)

In fact, by (A.18), we have maxik E|∑knj=1(u j (λ

∗ik

)− uj (λ∗ik

))|2 = O(n−c) for any c > 0.

Recall (A.12) for v j (λ). By independence, ‖∑kn+1j=1 v j (λ)‖2 = ∑kn+1

j=1 ‖v j (λ)‖2. We now

estimate ‖v j (λ)‖2. Let Yt,m(λ) = Xt,m ∑t−1s=t−4m+1 an,t−s Xs,m , Y ∗

t,m(λ) = Xt,m ∑t−4ms=1

an,t−s Xs,m . As in the proof of (A.2), routine calculations yield that, for 1 ≤ j ≤ kn ,

maxλ

‖v j (λ)‖2 ≤ 2maxλ

∥∥∥∥∥∥∑t∈Ij

Y ∗t,m(λ)

∥∥∥∥∥∥2

+2maxλ

∥∥∥∥∥∥∑t∈Ij

(Yt,m(λ)−EYt,m(λ))

∥∥∥∥∥∥2

= O(m B2n )+m ∑

t∈Ij

‖Yt,m(λ)‖2 = O(m B2n )

and maxλ ‖vkn+1(λ)‖2 = O(

m B2+2βn

), which, together with the fact that β,γ are suffi-

ciently small numbers, imply maxλE|∑kn+1j=1 v j (λ)|2 = O(nB1−ε

n ) for some ε > 0. So wehave

maxik E|∑knj=1 u j (λ

∗ik

)− gn,m(λ∗ik

)|2 = O(nB1−ε/2n ).

We next prove that

maxλE|gn,m(λ)− gn,m(λ)|2 = O(nBn/(log Bn)2). (A.23)


Let It = ∑t−ms=1 an,t−s(Xs,m − Xs,m). As in the proof of (A.2),

maxλ

∥∥∥∥∥ n

∑t=1

Xt,m It

∥∥∥∥∥2

≤ Cm maxλ

n

∑t=1

‖Xt,m‖2‖It‖2

= O(m2nBn)E(Xs,m − Xs,m)2 = O(m2(nBn)1−2α)

and, for Jt = ∑t−1s=t−m+1 an,t−s(Xs,m − Xs,m),

maxλ

∥∥∥∥∥ n

∑t=1

Xt,m Jt

∥∥∥∥∥2

≤ Cm3n‖Xs,m − Xs,m‖2 = O(m3n(nBn)−2α).

Hence maxλ ‖∑nt=1 Xt,m(It + Jt )‖ = O(τn). Similarly, we can also have ‖∑n

t=1(Xt,m −Xt,m)∑t−1

s=1 an,t−s Xs,m‖ = O(τn). Thus (A.23) is proved. By Proposition 1, maxλ ‖gn(λ)− gn,m(λ)‖ = O(τn). So to prove (A.22) it suffices to show that

|Cov(gn(λ∗ik

),gn(λ∗il ))| = O(τ2

n ),

which follows from Lemma A.5(i) immediately. Similarly, from Lemma A.5(iii) we have∣∣∣∣∣∣E[

kn

∑j=1

u j (λ∗ik

)

]2

−4π2nBn f 2(λ∗ii )

∣∣∣∣∣∣= O(τ2n ).

This together with (A.22) yields (A.21), and hence∣∣∣Cov1/2(Nn)−2π√

nBnId

∣∣∣= O(√

nBn/(log Bn)2)

. (A.24)

Let N be a standard normal Rd -valued random vector. By virtue of (A.24) it follows fromthe tail probabilities of a normal variable that

P

(∣∣∣Cov1/2(Nn)−2π√

nBnId ||N∣∣∣≥ τn

)= O(e−(log Bn)2/4),

which, together with (A.20), yields that

P

(|Vn |d ≥ yn

√nBn

)≤ P(

2π√

nBn |N |d ≥ yn√

nBn −2τn

)+ O

(e−(log Bn)2/4

)

= (1+o(1))

(√8π y−1

n exp

(− y2

n

8π2

))d

. (A.25)

Similarly, for (A.25) the reverse direction with ≥ also holds. Hence

P

(|Vn |d ≥ yn

√nBn

)= (1+o(1))

(√8π y−1

n exp

(− y2

n

8π2

))d

(A.26)

uniformly on {(λ∗i1

, . . . ,λ∗id

), (log Bn)2 ≤ i1 < · · · < id ≤ En, |λ∗i j

−λ∗ik

| ≥ (log Bn)2/Bn,

k �= j}.


By Lemma A.5(ii) and Lemma 2 in Berman (1962), similarly to (A.25) we have

P

(∣∣∣∣∣ kn

∑j=1

u j (λ∗ik

)

∣∣∣∣∣≥ yn√

nBn f (λ∗ik

), k = 1, . . . ,d

)

≤ C

(√8π y−1

n exp

(− y2

n

8π2

))d−2

× y−2n exp

(− y2

n

8π2 (1+ δ)

)(A.27)

for some δ > 0, uniformly on {λ∗i2

−λ∗i1

≥ B−1n ,λ∗

ik−λ∗

ik−1≥ B−1

n (log Bn)2; k = 3, . . . ,d}.The details of the derivation are omitted. Let tn = 2log Bn − log(π log Bn)+ x . Define

Ai =⎧⎨⎩ |∑kn

j=1 u j (λ∗i )|2

4π2nBn f 2(λ∗i )

≥ tn

⎫⎬⎭ and A = ⋃(log Bn)2≤i≤En

Ai .

By Bonferroni’s inequality, we have for every fixed k that

2k

∑t=1

(−1)t−1 Pt ≤ P(A) ≤2k−1

∑t=1

(−1)t−1 Pt ,

where Pt = ∑(log Bn)2≤i1<···<it ≤EnP(Ai1 ∩ ·· · ∩ Ait ). By (A.26) and (A.27), it follows

as in Woodroofe and Van Ness (1967) and Watson (1954) that Pt → exp(−t x/2)/t! as

n → ∞. Thus P(A) → 1− e−e−x/2, and the proof is complete. n

Proof of Theorems 3–5. From Lemmas A.1 and A.1∗, Remark A.2, and (A.16), weonly need to prove

P

(max

0≤i≤Bn

|gn,m(λ∗i )−Egn,m(λ∗

i )|24π2nBn f 2(λ∗

i )κ−2log Bn + log(π log Bn) ≤ x

)→ e−e−x/2

.

By Lemmas A.2 and A.3, we see that

max0≤i≤Bn

∣∣∣∣∣gn,m(λ∗i )−Egn,m(λ∗

i )−kn

∑j=1

u j (λ∗i )

∣∣∣∣∣= oP

(√nBn

log Bn

).

This together with Lemmas A.4 and A.6 implies the theorems. n

asymptotics of spectral density estimates - the

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