ASYMPTOTICS OF SPECTRAL DENSITY ESTIMATES

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

ASYMPTOTICS OF SPECTRAL DENSITY ESTIMATES WEIDONG LIU Zhejiang University

WEI BIAO WU University of Chicago

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

1. INTRODUCTION A fundamental problem in spectral analysis of time series is the estimation of spectral density functions. Let X k , k ∈ Z, be a stationary process with mean zero and finite covariance function γk = E(X 0 X k ). Assume that

∑ |γk | < ∞.

k∈Z

Let ı =

(1.1)

√

f (θ) =

−1 denote the imaginary unit. Under (1.1), the spectral density function

1 2π

1

∑ γk eıkθ = 2π ∑ γk cos(kθ),

k∈Z

0 ≤ θ < 2π,

(1.2)

k∈Z

exists and is continuous. The primary goal of the paper is to consider asymptotic properties of estimates of f . Based on observations X 1 , . . . , X n , let the sample covariances γˆk =

1 n ∑ X i X i−|k| , n i=|k|+1

1 − n ≤ k ≤ n − 1.

(1.3)

It is well known that the periodogram In (θ) =

1 1 n−1 |Sn (θ)|2 = ∑ γˆk eıkθ , 2πn 2πn k=1−n

where Sn (θ) =

n

∑ X k eıkθ ,

k=1

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

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c Cambridge University Press 2009

0266-4666/10 $15.00

ASYMPTOTICS OF SPECTRAL DENSITY ESTIMATES

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is an asymptotically unbiased but inconsistent estimate of f (θ). In the paper we consider the lag-window estimate f n (θ) =

1 2π

n−1

∑

K (k/Bn )γˆk eıkθ ,

(1.4)

k=1−n

where bn = Bn−1 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 has bounded 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 1 n ∑n X i γˇk = (X i − X¯ n )(X i−|k| − X¯ n ), where X¯ n = i=1 . ∑ n i=|k|+1 n A variant of (1.4) that allows unknown mean μ = EX j is the following estimate: 1 fˇn (θ) = 2π

n−1

∑

K (k/Bn )γˇk eıkθ .

(1.5)

k=1−n

Properties of spectral density estimates have been explored in many classical textbooks 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 references. It seems that many of the previous results require restrictive conditions on the underlying processes such as linear processes or strong mixing processes. Other contributions can be found in Phillips, Sun, and Jin (2006, 2007) and Velasco and Robinson (2001). In this paper we shall present an asymptotic theory for f n (θ) under very mild and natural conditions, thus substantially extending the applicability of spectral analysis to nonlinear and/or non–strong mixing processes. Some open problems are solved under our dependence framework (2.1). The rest of the paper is structured as follows. Main results are presented in Section 2 and proved in the Appendix. Section 3 provides bounds for approximations by m-dependent random variables, and Section 4 presents inequalities for 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 ∈ L p , 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 complex numbers. Denote by C p a constant that only depends on p and denote by C an absolute constant. Their values may vary from display to display. For two positive sequences (an ) and (bn ), write an ∼ bn if limn→∞ an /bn = 1 and an bn if, for some c > 0, c ≤ an /bn ≤ c−1 holds for all sufficiently large n. Define ω(u) ¯ = 2 if u/π ∈ Z and ω(u) ¯ = 1 if u/π ∈ Z.

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WEIDONG LIU AND WEI BIAO WU

2. MAIN RESULTS Assume throughout the paper that ε j , j ∈ Z, are independent and identically distributed (i.i.d.) random variables and R is a measurable function such that X j = R(. . . , ε j−1 , ε j ) = R(F j ),

where F j = (. . . , ε j−1 , ε j ),

(2.1)

is well defined. The class of processes under the framework (2.1) is huge; see Wiener (1958), Priestley (1988), Tong (1990), and Wu (2005), among others. Shao and Wu (2007) provide examples of nonlinear time series that are of form (2.1). To develop an asymptotic theory for the spectral density estimate f n (·), we need to introduce appropriate dependence measures. Following Wu (2005), we shall apply the idea of coupling and use a physical dependence measure. Let {ε j , εk : j, k ∈ Z} be i.i.d. random variables. For a set T ⊂ Z, let ε j,T = εj if j ∈ T , and ε j,T = ε j if j ∈ T . Let F j,k,T = (εl,T , j ≤ l ≤ k). If a random variable W is a function of F−∞,∞ , say, W = w(F−∞,∞ ), write WT = w(F−∞,∞,T ). For X j ∈ L p , p > 0, and T = {0}, define the physical dependence measure δ j, p = X j − X j,{0} p .

(2.2)

Here, by our convention, X j,{0} = R(F j,{0} ), where F j,{0} = (. . . , ε−1 , ε0 , ε1 , . . . , ε j ). Namely, X j,{0} is obtained by replacing ε0 in X j by ε0 . If j < 0, δ j, p = 0. If we view F j as input and X j as output of a physical system, then δ j, p measures the dependence of X j on the input ε0 via coupling. In many situations it is easy to work with δ j, p , which is directly related to the underlying data-generating mechanism (Wu, 2005). Example 1 ∞ Let X j = g(∑l=0 al ε j−l ), where al are real coefficients, εl are i.i.d. with εl ∈ L p , p > 0, and g is a Lipschitz continuous function. For j ≥ 0, we have X j,{0} = j−1 g(∑ l=0 al ε j−l + a j ε0 + ∑∞ l=1+ j al ε j−l ). Hence δ j, p = X j − X j,{0} p = O(|a j | ε0 − ε0 p ) = O(|a j |). 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 , ε j ), where ε j are i.i.d. and g is a measurable function. Assume that there exist p > 0 p and x0 ∈ R such that g(x0 , ε0 ) ∈ L p and E(L ε0 ) < 1, where L ε0 = supx=x |g (x, ε0 ) − g(x , ε0 )|/|x − x |. Then (X j ) has a unique stationary solution of the form (2.1), and δ j, 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-dependent exponential autoregressive processes, asymmetric generalized autoregressive conditionally heteroskedastic processes, and signed volatility models.


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Sections 2.1, 2.2, and 2.3 concern consistency, asymptotic normality, and maximum deviations of f n (·), respectively. Our results are all based on δ j, p . Define m, p =

∞

∞

∑ δj, p and m, p = ∑

j=m

p δ j, p

1/ p ,

where p = min(2, p).

(2.3)

j=m

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

2.1. Consistency To state our consistency result, we need some regularity conditions on the kernel K . Slightly different forms are needed for asymptotic normality and maximum deviations; see Conditions 2 and 3 in Sections 2.2 and 2.3. All those conditions on K are mild, and they are satisfied for Parzen, triangle, Tukey, and many other commonly 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) = ∞ ∞ 2 du =: κ < ∞ and its Fourier transform Kˆ (x) = −∞ K (u)eıxu du −∞ K (u) ∞ satisfies −∞ | Kˆ (x)| dx < ∞. THEOREM 1. Let Condition 1 be satisfied. Assume that E(X k ) = 0, X k ∈ L p , p ≥ 2, and 0, p = ∑∞ j=0 δ j, p < ∞. Let Bn → ∞ and Bn = o(n) as n → ∞. Then sup f n (θ) − f (θ) p/2 → 0.

θ ∈R

(2.4)

Because f n and f are even and have period 2π, supθ∈R in (2.4) is equivalent to supθ ∈[0,π ] . Remark 1. √ By Theorem 2 in Wu (2005), under 0, p < ∞, we have X 1 + · · · n X i−k p/2 ≤ X¯ n p + X j p = O( j ). Hence, for 0 ≤ k ≤ n − 1, X¯ n ∑i=1+k n ∑i=1+k X i−k p = O(1), from which, by elementary calculations, we obtain max|k|≤n−1 γˆk − γˇk p/2 = O(n −1 ). Assume n−1

∑

k=1−n

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

(2.5)

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Then (2.4) also holds if f n therein is replaced by fˇn in view of n−1 sup | f n (θ) − fˇn (θ)| ≤ 1 ∑ |K (k/Bn )| max γˆk − γˇk p/2 = O(Bn /n). |k|≤n 2π k=1−n θ∈R p/2 (2.6) With (2.6), Theorems 2–5 in Sections 2.2 and 2.3 also hold if f n therein is replaced by fˇn . Condition (2.5) holds for Epanechnikov, triangle, Parzen, and many other 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 density function. If 0,2 = ∞, then f may not exist. Consider, for example, the linear ∞ al ε j−l , where εl are i.i.d. with mean zero and variance 1. Then process X j = ∑l=0 √ ∞ ı jθ 2 δ j,2 = |a j |ε0 − ε0 = |a j | 2 and 2π f (θ) = | ∑∞ j=0 a j e | . If ∑ j=0 a j = ∞, for example, a j = 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-range dependent. Davidson and de Jong (2002) considered a closely related problem of estimating the variance sn2 = var(X 1 + · · · + X n ), in which X i are mean zero random vari∞ ables. They proved that, for the process X i = ∑∞ j=0 a j ηi− j , where ∑ j=0 |a j | < ∞ Bn and ηn is L 2 -near epoch dependent (NED) of size − 12 , sˆn2 = ∑nk=1 X k2 + 2 ∑k=1 2 2 K (k/Bn )n γˆk satisfies sˆn /sn → 1 in probability. Their result and our Theorem 1 have different ranges of applicability. Consider the case that both results are ap∞ ∞ 2 2 plicable: X i = ∑∞ j=0 a j ηi− j , where ηi = (∑ j=0 b j εi− j ) − ∑ j=0 b j and εi are i.i.d. N (0, 1). Their condition of L 2 NED of size − 12 requires that ηn − E(ηn |ε1 , . . . , εn ) = O(n −q ) for some q > 12 . With elementary calculations, the latter condition 2k+1 −1 2 −2q ), which implies that ∞ |b | ≤ ∞ is reduced to ∑∞ ∑ j=2 j ∑k=1 ∑ j=2k j=n b j = O(n 1 k/2 O(2−qk ) < ∞. We now apply our Theorem 1 with p = 2. b2j 2 2k/2 = ∑∞ k=1 2 n Elementary calculations show that physical dependence measure δn,2 = ∑i=0 ∞ O(|ai bn−i |). Hence our condition 0,2 < ∞ only requires ∑ j=0 |a j | < ∞ and ∑∞ j=0 |b j | < ∞. Hence in this example their NED-based condition is slightly stronger. Jansson (2002) considered the consistency of covariance matrix estimation for linear process with ηn being Rd -valued martingale differences. In the √ special 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 asymptotic distributional theory for the spectral density estimate f n (θ). With the latter results one can perform statistical inference such as hypothesis testing and construction of confidence intervals. However, it turns out that the central limit problem for


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f n (θ) is highly nontrivial. Earlier results require stringent conditions. The case of linear processes has been dealt with in Anderson (1971). Rosenblatt (1984) obtained a central limit theorem (CLT) under strong mixing and cumulant summability 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 f n (θ) under very mild and natural conditions, and it allows a wide class of nonlinear processes. In Theorem 2, the condition on dependence 0,4 < ∞ is natural, because otherwise the process (X j ) may be long-range dependent and the spectral density function may not be well defined. In Rosenblatt (1985), a summability condition of eighth-order joint cumulants is required. Rosenblatt asked whether the eighth-order summability condition can be weakened to the fourth order. The latter conjecture is solved in Theorem 2 in the sense that it imposes a summability condition of fourth-order physical dependence measures, which in many applications has the additional advantage that it is easier to work with than conditions on joint cumulants. The bandwidth condition Bn → ∞ and Bn = o(n) is also natural; see our consistency result, Theorem 1. Recall that ω(u) ¯ = 2 if u/π ∈ Z and ω(u) ¯ = 1 if u/π ∈ Z. Theorem 2 is proved in the Appendix. Condition 2. K is symmetric and bounded, limu→0 K (u) = K (0) = 1, and ∞ κ := −∞ K 2 (x) dx < ∞. Further assume that K is continuous at all but a finite number of points and suppose that sup0 0 such that, for all large n, c1 n δ ≤ Bn ≤ c2 n δ holds. Condition 5. −T1 (a) Let d m,q = ∑∞ t=0 min(δt,q , m+1,q

). Assume dn, p = O(n ) with T1 > max 12 − ( p − 4)/(2 pδ), 2δ/ p .

(b) n, p = O(n −T2 ), T2 > max[0, 1 − ( p − 4)/(2 pδ)]. THEOREM 3. Assume X 0 ∈ L p , p > max(4, 2/(1 − δ)), and EX 0 = 0. Further assume Conditions 3, 4, 5(a), and 5(b). Let λi∗ = π |i|/Bn . Then, for all x ∈ R,

n | f n (λi∗ ) − E f n (λi∗ )|2 −x/2 P max . − 2 log Bn + log(π log Bn ) ≤ x → e−e ∗ 2 0≤i≤Bn Bn f (λi )κ (2.8) Theorem 3 requires the moment condition X i ∈ L p with p > max(4, 2/(1 − δ)) → ∞ as δ → 1. Theorems 4 and 5 aim to weaken the latter moment condition. Conditions 3, 4, and THEOREM 4. Assume EX 0 = 0, X 0 ∈ L p , p > 4, and ∞ −iλx 5(a). Further assume that K is continuous and Kˆ (x) := −∞ e K (λ) dλ satis∞ ˆ fies −∞ | K (x)| dx < ∞. Then (2.8) holds. THEOREM 5. Assume Conditions 3 and 4 and EX 0 = 0, X 0 ∈ L p , p > 4. Further assume that δn, p = O(ρ n ) for some 0 < ρ < 1. Then (2.8) holds.


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Remark 4. Theorems 3–5 allow nonlinear processes. When they are applied to linear processes, our conditions are weaker than the classical one in Woodroofe and Van Ness (1967). To derive (2.8), the latter paper requires δ < 25 , δ ≥ 14 , ε0 ∈ L8 , and |ak | = O(k −1−β ) with β > 15 . If p = 8 and δ < 25 , our Theorem 4 √ only requires the weaker condition |ak | = O(k −1−β ) with β > ( 14 − 2)/10. We also 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 f n (θ) − f (θ) = O(Bn−1 ). Hence we can replace E f n (λi∗ ) in (2.8) by f (λi∗ ) if n log n = o(Bn3 ). 3. APPROXIMATIONS BY m-DEPENDENT PROCESSES With the physical dependence measure δ j, p in (2.2), we are able to provide explicit error bounds for approximating functionals of X k by functionals of the m-dependent process X˜ t := X t,m = E(X t |εt−m , . . . , εt ) = E(X t |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 this section is for quadratic forms. Proposition 2 in this section gives a martingale approximation for quadratic forms of m-dependent processes. LEMMA 1. Assume X i ∈ L p , p > 1, and E(X k ) = 0. Let C p = 18 p 3/2 ( p − 1)−1/2 and p = min(2, p). Let α1 , α2 , . . . , ∈ C. Then 1/ p n n p . (3.2) ∑ αk (X k − X˜ k ) ≤ C p An m+1, p , where An = ∑ |αk | k=1 k=1 p

Also, we have (i) ∑nk=1 αk X k p ≤ C p An 0, p and (ii) ∑nk=1 αk X˜ k p ≤ C p An 0, p . Proof. Let Dk, j := E(X k |Fk− j,k )−E(X k |Fk− j+1,k ). Then Dk, j , k = n, . . . , 1, form martingale differences with respect to Fk− j,∞ , and Dk, j p ≤ δ j, p . By Minkowski’s and Burkholder’s inequalities (see Wu and Shao, 2007, Lem. 1), we have p n n n p ∑ αk Dk, j ≤ C pp ∑ αk Dk, j pp ≤ C pp ∑ |αk | p δ j, p . k=1 k=1 k=1 p

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

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Because X k = ∑∞ j=0 Dk, j , the preceding argument implies (i). By Jensen’s inequality, δ˜k, p := X˜ k − X˜ k,{0} p = E(X k |Fk−m,k ) − E(X k,{0} |Fk−m,k,{0} ) p = E(X k − X k,{0} |Fk−m,k , ε0 ) p ≤ δk, p .

(3.3)

n

So (ii) follows from (i).

PROPOSITION 1. Assume EX 0 = 0, E|X 0 |2 p < ∞, p ≥ 2, and 0,2 p < ∞. Let Ln =

∑

1≤ j< j ≤n

αj − j X j X j

and

L˜ n =

∑

1≤ j< j ≤n

α j − j X˜ j X˜ j ,

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

L n − EL n − ( L˜ n − E L˜ n ) p 1 2

n An 0,2 p

≤ C p dm,2 p ,

∞

where dm,q = ∑ min(δt,q , m+1,q ). t=0

(3.4) t−1 ˜ ˜ ˜ Proof. Let Z t−1 = ∑t−1 j=1 αt− j X j , Z t−1 = ∑ j=1 αt− j X j , Yt = X t Z t−1 , Yt = X˜ t Z˜ t−1 , and

L n =

∑

1≤ j< j ≤n

α j − j X˜ j X j =

n

∑ X t Z˜ t−1 .

t=2

Recall that the notation X j,{k} represents a coupled version of X i = R(Fi ) by replacing εk in Fi by εk . If k > j, then X j,{k} = X j . We similarly define Z t−1,{k} and Z˜ t−1,{k} . So n Pk (L n − L n ) p ≤ ∑ X t (Z t−1 − Z˜ t−1 ) − X t,{k} (Z t−1,{k} − Z˜ t−1,{k} ) t=2 p n ≤ ∑ X t,{k} [(Z t−1 − Z˜ t−1 ) − (Z t−1,{k} − Z˜ t−1,{k} )] t=2 p

n

+ ∑ (X t − X t,{k} )(Z t−1 − Z˜ t−1 ) p =: Ik + I Ik .

(3.5)

t=2

By (3.3), X˜ j − X˜ j,{k} 2 p ≤ δ j−k,2 p . Because X j − X˜ j 2 p ≤ m+1,2 p , we have X j − X˜ j − X j,{k} + X˜ j,{k} 2 p ≤ 2 min(δ j−k,2 p , m+1,2 p ). By Lemma 1, n t−1 Ik = ∑ X t,{k} ∑ αt− j (X j − X˜ j − X j,{k} + X˜ j,{k} ) t=2 j=1 p


n−1 n = ∑ (X j − X˜ j − X j,{k} + X˜ j,{k} ) ∑ αt− j X t,{k} j=1 t=1+ j

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p

n−1

≤ 2 ∑ min(δ j−k,2 p , m+1,2 p )C2 p An 0,2 p . j=1

By Lemma 1, Z t−1 − Z˜ t−1 2 p ≤ C2 p An m+1,2 p . Because ∑nt=2 δt−k,2 p ≤ 0,2 p and (X t − X t,{k} )(Z t−1 − Z˜ t−1 ) p ≤ X t − X t,{k} 2 p Z t−1 − Z˜ t−1 2 p , n

∑

n

∑

I Ik2 ≤ C p A2n 2m+1,2 p

k=−∞

k=−∞

n

∑

0,2 p

n

∑

Ik2 ≤ C p A2n 20,2 p

k=−∞

∑ δj−k,2 p ≤ C p A2n n 2m+1,2 p 20,2 p ,

j=1

0,2 p

k=−∞

n−1

n−1

∑ min(δj−k,2 p , m+1,2 p )

j=1

≤ C p A2n 20,2 p n 0,2 p dm,2 p . Hence, by (3.5), because m+1,2 p ≤ dm,2 p , L n − EL n − (L n − EL n )2p ≤ C p

n

∑

2 Pk (L n − L n )2p ≤ C p n A2n 20,2 p dm,2 p,

k=−∞

which, by a similar inequality for L n − EL n − ( L˜ n − E L˜ n )2p , implies (3.4).

n

PROPOSITION 2. Assume EX 0 = 0, X 0 ∈ L4 , and 0,4 < ∞. Let α j = β j eı jλ , where λ ∈ R, β j ∈ R, 1 − n ≤ j ≤ −1, m ∈ N, and L˜ n = ∑1≤ j m. Also Dk are m-dependent, and they form martingale differences with respect to Fk . Let = eıλ , U j = j−t E(A j |F j−1 ), c4 = X 0 4 , and β j = 0 if j ≥ 0 or j ≤ −n. Observe that X˜ k = Ak − E(Ak+1 |Fk ) and A1 4 ≤ 2mc4 . Then t−8m t−8m ∑ α j−t ( X˜ j − D j ) = ∑ β j−t (U j − U j+1 ) j=1 j=1

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t−8m ≤ Cmc4 max |β j | + ∑ (β j−t − β j−1−t )U j . j j=1

(3.7)

m P j−l U j and P j−l U j , j ∈ Z, are martingale differences, Note that U j = ∑l=1

2 t−8m t−8m ∑ (β j−t − β j−1−t )P j−l U j = ∑ (β j−t − β j−1−t )2 P−l U0 2 . j=1 j=1 m m Because ∑l=1 P−l U0 ≤ m 1/2 (∑l=1 P−l U0 2 )1/2 ≤ Cm 3/2 c4 , by (3.7), t−8m ∑ α j−t ( X˜ j − D j ) ≤ C Vm1/2 ( β)mc4 . j=1

(3.8)

(3.9)

¯ similarly, we Let Q j = j−t E( A¯ j |F j−1 ). Using X˜ k = A¯ k − E( A¯ k+1 |Fk ), have n n ∑ α j−t ( X˜ t − D¯ t ) = ∑ β j−t (Q j − Q j+1 ) ≤ C Vm1/2 (β)mc4 . (3.10) t= j+8m t= j+8m j−t ( X ˜ j − D j ). Then W1,t , W1,t+4m , W1,t+8m , . . . are Let W1,t = X˜ t ∑t−8m j=1 β j−t 1/2

1/2

martingale differences. By (3.9), W1,t ≤ C Vm (β)mc42 . Write = Vm (β)m 3/2 n 1/2 c42 . So n 4m−1 (n−i)/(4m)

W1,i+4ml ≤ C. (3.11) ∑ W1,t ≤ ∑ ∑ t=1 i=1 l=0 j−t ( X ˜ j − D j ) are 12mLet Wt = W1,t + W2,t , where W2,t = X˜ t ∑t−1 j=t−8m+1 β j−t 1/2 2 dependent. As in (3.9), W2,t ≤ C Vm (β)mc4 . Similarly as (3.11), we have ∑nt=1 (W2,t − EW2,t ) ≤ C. By (3.11), ∑nt=1 (Wt − EWt ) ≤ C. Similarly, using (3.10), we have ∑nt=1 (Wt◦ −EWt◦) ≤ C for Wt◦ = (X˜ t − D¯ t) ∑t−1 j=1 α j−t D j . Hence Proposition 2 follows. n

4. INEQUALITIES FOR m-DEPENDENT PROCESSES As argued in Section 3, quadratic forms of (X k ) can be approximated by those of m-dependent random variables. So probability inequalities under m-dependence are useful for the asymptotic spectral estimation problem. Lemma 2, which follows, is an easy consequence of Corollary 1.6 in Nagaev (1979) via a simple blocking argument. We omit its proof. Proposition 3 is a Fuk–Nagaev-type inequality for quadratic forms of m-dependent random variables. It is useful for proving the maximum deviation results in Section 2.3.


1229

LEMMA 2. Let (X k ) be m-dependent with EX k = 0 and X k ∈ L p , p ≥ 2. Let n X i . Then for any Q > 0, there exists C1 , C2 > 0 only depending on Q Wn = ∑i=1 such that

m Q P(|Wn | ≥ x) ≤ C1 2 EWn2 x

n m p−1 n x +C1 min ∑ X k pp , ∑ P |X k | ≥ C2 m . x p k=1 k=1 PROPOSITION 3. Let (X t ) be m-dependent with EX t = 0, |X t | ≤ M a.s., t−1 m ≤ n, and M ≥ 1. Let Sk,l = ∑l+k t=l+1 X t ∑s=1 an,t−s X s , where l ≥ 0, l + k ≤ n and assume max1≤t≤n |an,t | ≤ K 0 , max1≤t≤n EX t2 ≤ K 0 , max1≤t≤n EX t4 ≤ K 0 for some K 0 > 0. Then for any x ≥ 1, y ≥ 1, and Q > 0, Q n

2 P(|Sk,l − ESk,l | ≥ x) ≤ 2e−y/4 + C1 n 3 M 2 x −2 y 2 m 3 (M 2 + k) ∑ an,s

+ C1 n 4 M 2 P |X 0 | ≥

C2 x 1

ym 2 (M + k 2 )

s=1

,

where C1 , C2 > 0 are constants depending only on Q and K 0 . Proof. Without loss of generality, we assume l = 0. Let Sk,0 = ∑kt=1 X t ∑t−2m s=1 an,t−s X s . Split the interval [1, k] into consecutive blocks H1 , . . . , Hkn with same size m. Here, for convenience, we assume kn = k/m ∈ N. Let Sj = ∑t∈Hj X t ∑t−2m s=1 an,t−s X s , 1 ≤ j ≤ kn . Then {Sj ; j = 1, 3, . . .} and {Sj ; j = 2, 4, . . .} are two sets of martingale differences. Let G j = σ ( Si ; 1 ≤ i ≤ j). By Freedman’s inequality (see Freedman, 1975), we have kn

P(| Sk,0 | ≥ 2x) ≤ 2e−y/4 + ∑ P(| Sj | ≥ x/y) j=1

kn + P ∑ E Sj I {Sj ≥ x/y}|G j−2 ≥ x j=1 kn 2 2 + P ∑ E[ S |G j−2 ] ≥ x /y =: 2e−y/4 + In + IIn + IIIn . j

j=1

n Because | Sj | ≤ CnmM2 , we have IIn ≤ C x −1 nmM 2 ∑kj=1 P(| Sj | ≥ x/y). By Lemma 2, t

x P | Sj | ≥ x/y ≤ m max P ∑ an,t−s X s ≥ s=1 ymM 1≤t≤n

1230


≤ C1 m x

−2 2

3

y m M

2

n

∑

Q 2 an,s

s=1

C2 x . + C1 mn P |X 0 | ≥ ym2 M

t2 −2m 1 −2m Because E( S 2j |G j−2 ) = ∑t1 ,t2 ∈Hj E(X t1 X t2 ) ∑ts=1 an,t−s X s ∑s=1 an,t−s X s , we have

IIIn ≤

2 2 P E[ S |G ] ≥ mx /(yk) j−2 ∑ j kn

j=1

t2 −2m t1 −2m x2 ≤ ∑ ∑ P ∑ an,t1 −s X s ∑ an,t2 −s X s ≥ ymk j=1 t1 ,t2 ∈Hj s=1 s=1 kn

t ≤ 2km max P ∑ an,t−s X s ≥ 1≤t≤n s=1

≤ C1 km x

−2

2

ym k

n

∑

Q 2 an,s

x 1

( ymk) 2

+ C1 kmnP |X 0 | ≥

s=1

C2 x 1

( ym3 k) 2

.

:= ∑kt=1 X t ∑t−1 It remains to consider Sk,0 s=t−2m+1 an,t−s X s . By Lemma 2, we have Q n C2 x −2 2 2 + C1 kmP |X 0 | ≥ 2 . P |Sk,0 − ESk,0 | ≥ x ≤ C1 x m ∑ at m M s=1

n

The proof is now complete. 5. A GENERAL CLT FOR QUADRATIC FORMS

In this section we establish a very general CLT for quadratic forms of stationary processes, which can be used for proving Theorem 2. For quadratic forms of independent random variables see de Jong (1987), Mikosch (1991), ten Vregelaar (1991), and Götze and Tikhomirov (1999), among others. For more references see Wu and Shao (2007), which gives a CLT for quadratic forms of martingale differences. 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 and ω(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

n

an, j− j X j X j

and

σn2 = ω(λ) ¯ ∑

n

2 . ∑ bn,t−k

k=1 t=1

(5.1)


1231

Assume that EX 0 = 0, E|X 0 |4 < ∞, 0,4 < ∞, and 2 = o(ςn2 ), max bn,t

where ςn2 =

0≤t≤n

n

2 ; ∑ bn,t

(5.2)

t=1

nςn2 = O(σn2 );

(5.3)

2 n ∑ ∑ ∑ an,k− j an,t− j = o(σn4 ); k=1 t=1 j=1+k n k−1

(5.4)

n

∑ |bn,k − bn,k−1 |2 = o(ςn2 ).

(5.5)

k=1

Then for 0 ≤ λ < 2π, σn−1 (Tn − ETn ) ⇒ N (0, 4π 2 f 2 (λ)). Proof. We shall apply Propositions 1 and 2 with α j = an, j = bn, j eı jλ . Recall Propositions 1 and 2 for L n , L˜ n , Dk , and Mn . Let L¯ n and M¯ n be the complex conjugates of L n and Mn . Note that Tn = L n + L¯ n +an,0 ∑nj=1 X 2j . Because E|X 0 |4 √ √ < ∞ and 0,4 < ∞, we have ∑nj=1 X 2j − nγ0 ≤ C nX 0 4 0,4 = O( n). Therefore, because dm,2 p → 0 as m → ∞, by (5.2), (5.3), and Propositions 1 and 2, it suffices to show that Mn + M¯ n ⇒ N (0, 4π 2 f˜2 (λ)), σn

where 2π f˜(λ) =

m

∑

γ˜j cos( jλ),

(5.6)

j=−m

holds for every m. Here γ˜j = E( X˜ 0 X˜ j ). Let Ut = ∑t−1 j=(t−4m+1)∨1 an, j−t D j . By √ n ¯ (5.2) and (5.3), we have ∑t=1 Dt Ut = O( n) max1≤t≤n |bn,t |. So it remains to show that 1 σn

n

∑

( D¯ t Ut + Dt U¯ t ) ⇒ N (0, 4π 2 f˜2 (λ)),

where Ut =

t=1+4m

t−4m

∑

an, j−t D j .

j=1

(5.7)

Because ∑nt=1+4m D¯ t Ut 44 = O(n)ςn4 = o(σn4 ), the Lindeberg condition easily follows. By the martingale CLT (see Hall and Heyde, 1980), (5.7) holds if n 1 E[( D¯ t Ut + Dt U¯ t )2 |Ft−1 ] → 4π 2 f˜2 (λ) ∑ σn2 t=1+4m

in probability

For the rest of the proof, we shall verify (5.8). Let −m ≤ l ≤ m − 1. Then 2 n n ∑ Pt+l ( D¯ t Ut + Dt U¯ t )2 = ∑ Pt+l ( D¯ t Ut + Dt U¯ t )2 2 t=1+4m t=1+4m ≤4

n

∑

Dt 44 Ut 44 .

t=1+4m

(5.8)

1232


Hence ∑nt=1+4m Pt+l ( D¯ t Ut + Dt U¯ t )2 = o(σn2 ). Because Dt is Ft−m,t measurable and Ut is Ft−3m measurable, E( D¯ t2 Ut2 |Ft−m−1 ) = Ut2 E( D¯ t2 ), (5.8) is then reduced to n 1 [U 2 E( D¯ t2 ) + U¯ t2 E(Dt2 ) + 2|Ut |2 E(|Dt2 |)] → D0 4 ∑ 2 σn t=1+4m t

in probability (5.9)

ı jλ = 2π f˜(λ). For the rest of the proof we by noting that D0 2 = ∑m j=−m γ˜j e shall only deal with the case of λ = 0, π because the case of λ = 0, π can be similarly proved. Using the argument of Lemma 3 in Wu and Shao (2007), under (5.2)–(5.4), we can prove that ∑nt=1+4m (Ut2 −EUt2 ) = o(σn2 ). So (5.9) is further reduced to n 1 [E(Ut2 )E( D¯ t2 ) + E(U¯ t2 )E(Dt2 ) + 2E(|Ut |2 )E(|Dt2 |)] → D0 4 . (5.10) ∑ σn2 t=1+4m 2 2 Clearly E(Ut2 ) = ∑t−4m j=1 an, j−t ED j . By summation by parts and (5.5), because j

n 2 2 | ∑l=1 e2ılλ | ≤ 1/| sin λ|, we have maxt≤n | ∑tj=1 an, j−t | = o(ςn ) and ∑t=1+4m |E(Ut2 )| = o(σn2 ). So (5.10) follows from (5.2) and (5.3) because E(|Ut |2 ) = t−4m 2 2 n ∑ j=1 bn, j−t E(|Dt |).

REFERENCES Anderson, T.W. (1971) The Statistical Analysis of Time Series. Wiley. Andrews, D.W.K. (1984) Nonstrong mixing autoregressive processes. Journal of Applied Probability 21, 930–934. Bentkus, R.Y. & R.A. Rudzkis (1982) On the distribution of some statistical estimates of spectral density. Theory of Probability and Its Applications 27, 795–814. Berman, S. (1962) A law of large numbers for the maximum of a stationary Gaussian sequence. Annals of Mathematical Statistics 33, 93–97. Brillinger, D.R. (1969) Asymptotic properties of spectral estimates of second order. Biometrika 56, 375–390. Brillinger, D.R. (1975) Time Series: Data Analysis and Theory. Holden-Day. Brockwell, P.J. & R.A. Davis (1991) Time Series: Theory and Methods, 2nd ed. Springer-Verlag. Bühlmann, P. and H.R. Künsch (1999) Block length selection in the bootstrap for time series. Computational Statistics and Data Analysis 31, 295–310. Davidson, J. & R.M. de Jong (2002) Consistency of kernel variance estimators for sums of semiparametric linear processes. Econometrics Journal 5, 160–175. de Jong, P. (1987) A central limit theorem for generalized quadratic forms. Probability Theory and Related Fields 75, 261–277. Einmahl, U. & D.M. Mason (1997) Gaussian approximation of local empirical processes indexed by functions. Probability Theory and Related Fields 107, 283–311. Freedman, D. (1975) On tail probabilities for martingales. Annals of Probability 3, 100–118. Götze, F. & A.N. Tikhomirov (1999) Asymptotic distribution of quadratic forms. Annals of Probability 27, 1072–1098. Grenander, U. & M. Rosenblatt (1957) Statistical Analysis of Stationary Time Series. Wiley. Hall, P. & C.C. Heyde (1980) Martingale Limit Theory and Its Applications. Academic Press.


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Jansson, M. (2002) Consistent covariance matrix estimation for linear processes. Econometric Theory 18, 1449–1459. Mikosch, T. (1991) Functional limit theorems for random quadratic forms. Stochastic Processes and Their Applications 37, 81–98. Nagaev, S.V. (1979) Large deviations of independent random variables. Annals of Probability 7, 745–789. Phillips, P.C.B., Y. Sun, & S. Jin (2006) Spectral density estimation and robust hypothesis testing using steep origin kernels without truncation. International Economic Review 47, 837–894. Phillips, P.C.B., Y. Sun, & S. Jin (2007) Long run variance estimation and robust regression testing using sharp origin kernels with no truncation. Journal of Statistical Planning and Inference 137, 985–1023. Politis, D.N., J.P. Romano, & M. Wolf (1999) Subsampling. Springer-Verlag. Priestley, M.B. (1981) Spectral Analysis and Time Series 1. Academic Press. Priestley, M.B. (1988) Nonlinear and Nonstationary Time Series Analysis. Academic Press. Rosenblatt, M. (1984) Asymptotic normality, strong mixing, and spectral density estimates. Annals of Probability 12, 1167–1180. Rosenblatt, M. (1985) Stationary Sequences and Random Fields. Birkhäuser. Rudzkis, R. (1985) On the distribution of the maximum deviation of the Gaussian stationary time series spectral density estimate. Lithuanian Mathematical Journal 25, 118–130. Shao, X. & W.B. Wu (2007) Asymptotic spectral theory for nonlinear time series. Annals of Statistics 35, 1773–1801. Song, W.M. & B.W. Schmeiser (1995) Optimal mean-squared-error batch sizes. Management Science 41, 110–123. ten Vregelaar, J.M. (1991) Note on the convergence to normality of quadratic forms in independent variables. Theory of Probability and Its Applications 35, 177–179. Tong, H. (1990) Non-linear Time Series: A Dynamical System Approach. Oxford University Press. Velasco, C. & P.M. Robinson (2001) Edgeworth expansions for spectral density estimates and Studentized sample mean. Econometric Theory 17, 497–539. Watson, G.S. (1954) Extreme values in samples from m-dependent, stationary, stochastic processes. Annals of Mathematical Statistics 25, 798–800. Wiener, N. (1958) Nonlinear Problems in Random Theory. MIT Press. Woodroofe, M. & J.W. Van Ness (1967) The maximum deviation of sample spectral densities. Annals of Mathematical Statistics 38, 1558–1569. Wu, W.B. (2005) Nonlinear system theory: Another look at dependence. Proceedings of the National Academy of Sciences 102, 14150–14154. Wu, W.B. & X. Shao (2007) A limit theorem for quadratic forms and its applications. Econometric Theory 23, 930–951.

TECHNICAL APPENDIX AND PROOFS A.1. Proofs of the Results in Sections 2.1–2.3. Proof of Theorem 1. Recall (3.1) for X˜ t = X t,m , which are m-dependent. Let an,t = n ˜ K (t/Bn ) cos(tθ). For 4m + 1 ≤ t ≤ n let Yt = X˜ t ∑t−4m s=1 an,t−s X s and Rm = ∑t=1+4m Yt . 2 By Condition 1, ∑s∈Z an,s = O(Bn ). By independence and Lemma 1, 1/2 t−4m t−4m 1/2 2 Yt p = X˜ t p ∑ an,t−s X˜ s ≤ X 0 p C p 0, p ∑ an,t−s = O(Bn ). (A.1) s=1 s=1 p

1234


Let Jl = (n − l)/(4m) . Because Yt , Yt+4m , Yt+8m , . . ., are L p martingale differences, 4m Jl 4m 1/2 1/2 Rn p ≤ ∑ ∑ Yl+4m j = ∑ Jl O(Bn ) = O[(mn Bn )1/2 ). (A.2) l=1 j=1 l=1 p

Let γ˜k = E( X˜ 0 X˜ k ), g˜ n (θ) = 2π fñ (θ) and g˜ n (θ) =

=

1 n ˜ 2 2 n ˜ t−1 ∑ X t + n ∑ X t ∑ an,t−s X˜ s n t=1 t=2 s=1 t−1 1 n ˜2 2 n ˜ 2Rn an,t−s X˜ s + . Xt + ∑ Xt ∑ ∑ n t=1 n t=2 s=max(1,t−4m+1) n

(A.3)

By the ergodic theorem, for 1 ≤ l ≤ 4m, n −1 ∑nt=1 X˜ t X˜ t+l − γ˜l p/2 → 0. So lim g˜ n (θ) − 2n −1 Rn − E[g˜ n (θ) − 2n −1 Rn ] p/2 = 0.

n→∞

(A.4)

Let Iñ (u) = (2π n)−1 | Sñ (u)|2 , where Sñ (u) = ∑nk=1 X˜ k eıku . By Lemma 1, Sn (u) − Sñ (u) p = O(n 1/2 ) m, p , Sn (u) p + Sñ (u) p = O(n 1/2 ). Hence In (u)− Iñ (u) p/2 = O(1) m, p . Because fñ (θ) = R Kˆ (u) Iñ (Bn−1 u + θ) du, p/2 ≥ 1, and R | Kˆ (u)| du < ∞, f n (θ) − fñ (θ) p/2 ≤

R

| Kˆ (u)|In (Bn−1 u + θ) − Iñ Bn−1 u + θ p/2 du = O(1) m, p .

So |E[ f n (θ) − fñ (θ)]| = O( m, p ). By (A.2) and (A.4), because Bn = o(n), limn→∞ fñ (θ) − E[ fñ (θ)] p/2 = 0. Hence limn→∞ f n (θ) − E f n (θ) p/2 = 0 in view of f n (θ) − E f n (θ) p/2 ≤ f n (θ) − fñ (θ) p/2 + |E[ f n (θ) − fñ (θ)]| + fñ (θ) − E[ fñ (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 f n (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 ) ∼ Bn κ and ∑nk=1 2 ∼ n Bn κ. So (5.2) and (5.3) hold. Let M be a positive integer that will be ∑nt=1 bn,t−k specialized later. Because Bn = o(n), by Schwarz’s inequality,

k−1

n

∑

∑

k=1 t=1∨(k−M Bn )

2

n

∑

bn,k− j bn,t− j

= O(n Bn3 ) = o(σn4 ).

(A.5)

j=1+k

Using Schwarz’s inequality again, we can get n k−M Bn

∑ ∑

k=1

t=1

2

n

∑

j=1+k

bn,k− j bn,t− j

≤ Cn 2 Bn

n

∑

k=M Bn

K 2 (k/Bn ) ≤ τ M n 2 Bn2

(A.6)


1235

with τ M → 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 obtained that |K (k/Bn ) − K ((k − 1)/Bn )| = o(1) uniformly for |k| ≤ M Bn , except for ε Bn MB points k ∈ [−M Bn , M Bn ], where ε > 0 is an arbitrary number. Thus, ∑k=1n {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)/

n

Bn ]}2 ≤ τ M Bn . Hence (5.5) is proved.

Remark A.1. It is easily seen that Theorem 2 also holds if the requirement that K is continuous at all but a finite number of points in Condition 2 is replaced by the following one: 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 define Tn,m by replacing {X t } in (5.1) by {X t,m }. Let gn (λ) = Tn − ETn − ∑nk=1 (X k2 − EX k2 ) 2 − EX 2 ). Then 2πn{ f (λ) − E f (λ)} = and gn,m (λ) = Tn,m − ETn,m − ∑nk=1 (X k,m n n k,m gn (λ) + ∑nk=1 (X k2 − EX k2 ). The proofs of Theorems 3–5 are quite complicated, and they √ are based on a series of lemmas. Let τn = n Bn /log Bn . LEMMA A.1. Suppose that the conditions of Theorem 3 hold. Then for any 0 < C < 1 there exists γ ∈ (0, C) such that, for m = n γ ,

max |gn (λi∗ ) − gn,m (λi∗ )| = oP n Bn / log Bn . 0≤i≤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 , 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, E|gn (λi∗ ) − gn,s1 (λi∗ )| p/2 P max |gn (λi∗ ) − gn,s1 (λi∗ )| ≥ τn ≤ (1 + Bn ) max 0≤i≤Bn 0≤i≤Bn (n Bn ) p/4 (log Bn )− p/2 l

p/2

≤ C Bn ds1 , p (log Bn ) p/2 ≤ Cn δ−T1 p/2 (log n) p/4−2 = o(1). So we only need to show that, for every 1 ≤ l ≤ r − 1, max0≤i≤Bn |gn,sl (λi∗ ) − gn,sl+1 √ (λi∗ )| = oP n Bn / log Bn . Let Yt,m (λ) = X t,m ∑t−1 s=1 X s,m an,t−s and note that, for any 1 ≤ l ≤ r , Yt,sl (λ), 1 ≤ t ≤ n, are (Bn + sl )-dependent. Split the interval [1, n] into consecutive blocks H1 , H2 , . . ., Htn with equal length Bn + sl , where the number of intervals tn ∼ n/(Bn + sl ) and the last interval may be incomplete. For convenience we assume that the length of the last interval is also Bn + sl . Define u˘ j (λ) =

∑ (Yt,sl (λ) − Yt,sl+1 (λ)),

u j (λ) = u˘ j (λ) − Eu˘ j (λ),

1 ≤ j ≤ tn . (A.7)

t∈Hj

t

n Then u 1 (λ), . . . , u tn (λ) are 1-dependent and gn,sl (λ) − gn,sl+1 (λ) = 2 ∑ j=1 u j (λ). By Lemma 2, for any large Q and 1 ≤ l ≤ r − 1, ∗ ∗ P max |gn,sl (λi ) − gn,sl+1 (λi )| ≥ τn

0≤i≤Bn

1236

WEIDONG LIU AND WEI BIAO WU Bn

∑

≤C

tn Q ∑ j=1 E|u j (λi∗ )|2 τn2

i=0

+C

Bn tn

∑ ∑P

|u j (λi∗ )| ≥ C Q τn .

(A.8)

i=0 j=1

Because dn, p = O(n −T1 ), by Proposition 1, maxλ∈R max1≤ j≤tn E|u j (λ)|2 = O Bn (Bn + −2T sl )sl+1 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 l 1 ≤ l ≤ r0 (n) − 1. Because sl ∼ sl+1 n ρ (1−ρ) and sr0 (n)−1 ≥ Bn , we have tn n/sl . By Proposition 1 and Markov’s inequality, Bn tn

∑ ∑P

i=0 j=1

p/4 − pT /2 |u j (λi∗ )| ≥ C Q τn ≤ C Bn nsl−1 sl sl+1 1 n − p/4 (log Bn ) p/2 p/4−1− pT1 /2 1− p/4+ε n (log Bn ) p/2 =: F1 ,

≤ C Bn sl

(A.9)

where ε → 0 as → 1. If 4 < p ≤ 4 + 4δ, then p/4 − 1 − pT1 /2 ≤ 0, and hence p/4− pT1 /2 1− p/4+ε n (log Bn ) p/2 ≤ Cn 0∨{δ( p/4− pT1 /2)}+1− p/4+ε (log Bn ) p/2

F1 ≤ C Bn

= 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, we have F1 ≤ Cn δ− 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

∑

Yˇt,sl = X t,sl Ut,l − X t,sl+1 Ut,l+1 .

an,t−s X s,s j ,

s=1

Let j1 = min{k : k ∈ Hj } and j2 = max{k : k ∈ Hj }. Because X t,i = ∑tk=t−sl Pk X t,i for i = sl , sl+1 , we have u j (λ) : =

∑

Yˇt,sl =

t∈Hj

=

(k+sl )∧ j2

∑

j2

∑

k= j1 −sl

W k,l ,

where W k,l

(Pk X t,sl Ut,l − Pk X t,sl+1 Ut,l+1 ).

t=k∨ j1

Note that W k,l , j1 − sl ≤ k ≤ j2 , are martingale differences; by Lemma 1, max W k,l p ≤ max λ∈R

(k+sl )∧ j2

∑

λ∈R t=k∨ j 1

+ max

Pk X t,sl p Ut,l − Ut,l+1 p

(k+sl )∧ j2

∑

λ∈R t=k∨ j 1

1/2

Pk (X t,sl − X t,sl+1 ) p Ut,l+1 p = O(Bn ).


1237

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

∑P 0≤i≤Bn

Bn max

|u j (λi∗ )| ≥ τn ≤ Cn 1− p/2 Bn−1 (log Bn )2 p

j=1

p/2

≤ Cn 1− p/2 Bn

j2

∑

max E|W k,l | p

k= j1 −sl λ∈R

(log Bn )2 p = o(1).

(A.10) ˇ Putting Yt,sl = Yt,sl (λ) − Yt,sl+1 (λ) − Yt,sl , then u j (λ) = u j (λ) + ∑t∈Hj (Yt,sl − EYt,sl ).

p/4 −T p/2 By Lemma 1, we have E|Yt,sl | p/2 ≤ Csl sl+12 . Because Yt,sl , 1 ≤ t ≤ n, are sl dependent, by Lemma 2, for Q large enough, tn

Bn max ∑ P |u j (λi∗ ) − u j (λi∗ )| ≥ τn 0≤i≤Bn j=1

≤ C Bn max

tn

∑

0≤i≤Bn j=1

sl ∑t∈Hj E|Yt,sl |2

Q

n Bn (log Bn )−2

+ C Bn (n Bn )− p/4 (log Bn ) p/2 sl

p/2−1

tn

∑ ∑ 0≤i≤Bn max

E|Yt,sl | p/2

j=1 t∈Hj

3 p/4−1 − pT2 /2 sl+1 (n Bn )− p/4+1 (log Bn ) p/2 = o(1),

≤ Csl

where the last relation is due to sl < Bn , sl ∼ sl+1 n (A.8)–(A.11), Lemma A.1 follows.

ρ l (1−ρ)

(A.11) and Condition 5(b). By

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 every r0 (n) ≤ l ≤ r − 1, max0≤i≤Bn |gn,sl (λi∗ ) − gn,sl+1 (λi∗ )| = oP n Bn / log Bn . Recall √ (A.7) for u j (λ). Let u j (λ) = u j (λ)I |u j (λ)| ≤ n Bn /(log Bn )3 . Then √ tn tn n Bn ∗ P max ∑ u j (λi ) ≥ τn ≤ P(G n ≥ τn ) + ∑ P Pj ≥ , 0≤i≤Bn j=1 (log Bn )3 j=1 n u j (λi∗ )| and Pj = max0≤i≤Bn |u j (λi∗ )|. By Proposition 1, where G n = max0≤i≤Bn | ∑ j=1

t

tn E|u j (λ)|2 ≤ Cn Bn ds2 ,4 , and ∑ j=1 l tn

tn

j=1

j=1

∑ |Eu j (λ)| ≤ ∑ E|u j (λ)|2 (n Bn )−1/2 (log Bn )3 ≤ C(n Bn )1/2 ds2l ,4 (log Bn )3 = o

n Bn / log Bn ,

we have, by Bernstein’s inequality, P(G n ≥ τn ) ≤ C Bn exp(−C(log Bn )2 ) = o(1). To finish the proof of Lemma A.1∗ , we only need to show that tn n := ∑ P Pj ≥ n Bn /(log Bn )3 = o(1). j=1

1238

WEIDONG LIU AND WEI BIAO WU D

Recall Yt,m (λ) = X t,m ∑t−1 s=1 X s,m an,t−s . Because for 2 ≤ j ≤ tn , u j (λ) = u 2 (λ) and 2Bn +2sl

∑

u˘ 2 (λ) =

(Yt,sl (λ) − Yt,sl+1 (λ)) −

Bn +sl

t=1

∑

(Yt,sl (λ) − Yt,sl+1 (λ)).

t=1 2B +2s

2B +2s

n n l l Let Sn,1 (u) = ∑k=1 X k,sl eıku and Sn,2 (u) = ∑k=1 X k,sl+1 eıku , and similarly de fine Sn,1 (u), Sn,2 (u), by replacing 2Bn + 2sl in Sn,1 (u) and Sn,2 (u) by Bn + sl . Then

2u˘ 2 (λ) = Bn

∞ −∞

∞

− Bn −

Kˆ (Bn (u − λ)) |Sn,1 (u)|2 − |Sn,2 (u)|2 du

−∞

(u)|2 − |S (u)|2 du Kˆ (Bn (u − λ)) |Sn,1 n,2

2Bn +2sl

∑

t=Bn +sl +1

2 + X t,s l

2Bn +2sl

∑

t=Bn +sl +1

2 X t,s . l+1

(u)|2 − E|S (u)|2 , k = 1, 2, Let Q n,k (u) = |Sn,k (u)|2 − E|Sn,k (u)|2 , Q n,k (u) = |Sn,k n,k Wn = maxu∈R |Q n,1 (u) − Q n,2 (u)|, and Wn = maxu∈R |Q n,1 (u) − Q n,2 (u)|. Then

2 max |u 2 (λ)| ≤ (Wn + Wn ) λ∈R

−∞

Define μn =

n

l

l

√

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

tn

tn

∑ P(Wn ≥ 2−1 μn ) + ∑ P(Wn ≥ 2−1 μn ) + o(1)

j=1

≤

| Kˆ (u)| du

2Bn +2sl 2Bn +2sl 2 2 2 2 + (X t,sl − EX t,sl ) + (X t,sl+1 − EX t,sl+1 ) . ∑ ∑ t=B +s +1 t=B +s +1 n

n ≤

∞

j=1

tn

∑P

j=1

+

tn

max (|Sn,1 (u)| + |Sn,2 (u)|) ≥ νn

0≤u≤2π

∑P

j=1

(u)| + |S (u)|) ≥ ν max (|Sn,1 n n,2 0≤u≤2π

+ o(1).

Let li = i/n 2 , 0 ≤ i ≤ 2π n 2 . Then max |Sn,1 (u)| ≤

0≤u≤2π

max

0≤i≤2πn 2

|Sn,1 (li )| + Cn −2 Bn

Bn +sl

∑

|X k,sl |.

k=1

t

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

Sn,1 (u) =

2Bn +2sl (2Bn +2sl )∧( j+sl )

∑

j=1−sl

∑

k=1∨ j

P j (X k,sl )eıku =:

2Bn +2sl

∑

j=1−sl

Rn, j (u).


1239

Set Rn, j (u) = Rn, j (u)I |Rn, j (u)| ≤ (n Bn )1/4 /(log Bn )5 . Then 2Bn +2sl max ∑ Rn, j (u) − R n, j (u) ≥ νn ∑ P 0≤u≤2π j=1−s j=1 l tn

4 (B +s )∧( j+sl ) (log Bn )18 n(log Bn )18 2Bn +2sl n l P j (X k,s ) = o(1), ≤C ≤C ∑ ∑ l Bn (n Bn ) j=1−s Bn k=1∨ j l

4

where R n, j (u) = Rn, j (u) − E( Rn, j (u)|F j−1 ). Also, 2Bn +2sl 2 max ∑ E(|R n, j (u)| |F j−1 ) u∈R j=1−s l

p/2

2Bn +2sl 2 ≤ max ∑ E(|Rn, j (u)| |F j−1 ) u∈R j=1−s l

≤

2Bn +2sl

∑

(2Bn +2sl )∧( j+sl )

∑

j=1−sl

p/2

2 δk− j, p

≤ C Bn .

k=1∨ j

We can get, by Freedman’s inequality and because p > 4, Bn +sl tn max ∑ R n, j (li ) ≥ νn ∑ P 0≤i≤2π n 2 j=1−s j=1 l

≤ Cn 3 Bn−1 exp(−C(log Bn )2 ) + n Bn−1 Bn

p/2

(n Bn )− p/4 (log Bn )5 p/2 = o(1).

n

The proof is now complete.

For Lemmas A.2–A.4, we need to introduce truncation. Let α < 14 be close to 14 sufficiently and m = n γ , where γ is small enough. Define =X α X t,m t,m I {|X t,m | ≤ (n Bn ) },

− EX , X t,m = X t,m t,m

n

t−1

n

t−1

t=2

s=1

t=2

s=1

g n,m (λ) = 2 ∑ X t,m

∑ an,t−s X s,m − 2E ∑ X t,m ∑ an,t−s X s,m .

1+β

Let pn = Bn , 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 I j by Hj = [( j − 1)( pn + qn ) + 1, j pn + ( j − 1)qn ]; I j = [ j pn + ( j − 1)qn + 1, j ( pn + qn )]; 1 ≤ j ≤ kn ,

Ikn +1 = [kn ( pn + qn ) + 1, n].

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

∑ (Y t,m (λ) − EY t,m (λ)

t∈Hj

and

v j (λ) =

∑ (Y t,m (λ) − EY t,m (λ)).

t∈I j

(A.12)

1240


Because K (·) is bounded, we have n E max gn,m (λ) − g n,m (λ) ≤ CE ∑ |X t,m | λ∈R

t=2

t−1

∑

s=1∨(t−Bn )

n

t−1

t=2

s=1∨(t−Bn )

+ CE ∑ |X t,m − X t,m | Recall τn =

n

∑

E

∑

(A.13)

|X s,m |.

√

n Bn /log Bn . By independence and because X 0 ∈ L p , p > 4,

|X t,m |

t=m+1

t−1

∑

s=1∨(t−Bn )

n

∑

≤E

|X t,m |

t=m+1

+E

|X s,m − X s,m |

|X s,m − X s,m |

t−m

∑

s=1∨(t−Bn )

|X s,m − X s,m |

n

t−1

t=2

s=(t−m+1)∨1

∑ |X t,m |

(A.14)

∑

|X s,m − X s,m |

≤ C(n Bn )1−( p−1)α + Cnm(n Bn )−( p−2)α = o(τn ). Similar arguments yield to E

n

t−1

t=2

s=1∨(t−Bn )

∑ |X t,m − X t,m |

∑

|X s,m |

= o(τn ).

(A.15)

Combining (A.13)–(A.15), we get E max gn,m (λ) − g n,m (λ) = o(τn ).

(A.16)

λ∈R

LEMMA A.2. Assume EX 0 = 0 and EX 04 < ∞. Recall τn = max |i | = OP (τn ) ,

|i|≤Bn

where i =

kn +1

∑

√

n Bn / log Bn . We have

v j (λi∗ ).

j=1

Proof. Because v j (λi∗ ), 1 ≤ j ≤ kn + 1, are independent, by Lemma 2, for all large Q, ⎛ P (|i | ≥ τn ) ≤ C ⎝

k +1

n Ev 2j (λi∗ ) ∑ j=1

⎞Q

kn +1 ⎠ + C ∑ P |v j (λ∗ )| ≥ C Q τn . i n Bn (log Bn )−2 j=1

In Proposition 3 we let x = C Q τn , M = (n Bn )α , k = Bn +m, m = n γ , and y = (log Bn )2 . Then for all c > 0, P(|v j (λi∗ )| ≥ C Q τn ) = O(n −c ). Because P(max|i|≤Bn |i | ≤ τn ) ≤ n ∑|i|≤Bn P (|i | ≥ τn ), by elementary manipulations, the lemma follows.


1241

LEMMA A.3. Assume that EX 0 = 0 and EX 04 < ∞. For 1 ≤ j ≤ kn let √ √ n Bn n Bn u j (λ) ≤ − Eu uj (λ) = u j (λ)I u j (λ) ≤ (λ)I j (log Bn )4 (log Bn )4

(A.17)

be the truncated version of u j (λ). Then kn ∗ ∗ −1 = o(1). P max ∑ (u j (λi ) − uj (λi )) ≥ n Bn (log Bn ) 0≤i≤Bn j=1

Proof. As in the proof of Lemma A.2, we can get √ n Bn P |u j (λi∗ )| ≥ C Q = O(n −c ) (log Bn )4

(A.18)

n

for any large c > 0. The lemma immediately follows. LEMMA A.4. Assume that EX 0 = 0 and EX 04 < ∞. We have for any x > 0, kn P max ∑ uj (λi∗ ) ≥ x n Bn log Bn = o(1), i∈B j=1 where B = {|i| ≤ (log Bn )2 } ∪ {Bn − (log Bn )2 ≤ |i| ≤ Bn }.

n

Proof. The lemma easily follows from Bernstein’s inequality. LEMMA A.5. Suppose that EX 0 = 0, EX 04 < ∞, and dn,4 = O((log n)−2 ). (i) We have |E[gn (λ1 ) − Egn (λ1 )][gn (λ2 ) − Egn (λ2 )]| ≤ Cn Bn (log Bn )−2

uniformly on {(λ1 , λ2 ) : 0 ≤ λk ≤ π − Bn−1 (log Bn )2 , k = 1, 2, and |λ1 − λ2 | ≥ Bn−1 (log Bn )2 }. (ii) For αn > 0 with lim supn→∞ αn < 1, we have, uniformly on {(λ1 , λ2 ) : Bn−1 (log Bn )2 ≤ λk ≤ π − Bn−1 (log Bn )2 , k = 1, 2, and |λ1 − λ2 | ≥ Bn−1 }, that |E[gn (λ1 ) − Egn (λ1 )][gn (λ2 ) − Egn (λ2 )]| ≤ 4π 2 αn n Bn f (λ1 ) f (λ2 )κ. (iii) We have uniformly on {Bn−1 (log Bn )2 ≤ λ ≤ π − Bn−1 (log Bn )2 } that |E[gn (λ) − Egn (λ)]2 − 4π 2 n Bn f 2 (λ)κ| ≤ Cn Bn (log Bn )−2 . Proof. (i) Let m = n with > 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 D¯ k,λ . Let Mn (λ) = ∑nt=1 D¯ t,λ ∑t−1 j=1 an, j−t D j,λ . By Propositions 1 and 2, it suffices to verify that rn,λ1 ,λ2 := |E[Mn (λ1 ) + M¯ n (λ1 )][Mn (λ2 ) + M¯ n (λ2 )]| ≤ C

n Bn . (log Bn )2

1242

WEIDONG LIU AND WEI BIAO WU an, j−t D j,λ . Because Dt,λ , t ≥ 1 are martingale Let Nn (λ) = ∑nt=1 D¯ t,λ ∑t−m−1 j=1 differences, elementary manipulations of trigonometric identities show that n t−m−1

r¯n,λ1 ,λ2 = 2|E D¯ 0,λ1 D¯ 0,λ2 |2

∑ ∑

K 2 ((t − s)/Bn ) cos((t − s)(λ1 + λ2 ))

t=1 s=1

+2|ED 0,λ1 D0,λ2 |2

n t−m−1

∑ ∑

K 2 ((t − s)/Bn )

t=1 s=1

× 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 summation by parts formula and Condition 3, it follows that n t−m−1 2 ∑ ∑ K ((t − s)/Bn ) cos((t − s)(λ1 ± λ2 )) t=1 s=1 ≤ Cnm + CB2n +

Bn 2 ∑ ∑ K (s/Bn ) cos s(λ1 ± λ2 ) t=B +m+1 s=1 n

n

≤ Cnm + CB2n + CnBn /(log Bn )2 , M which by (A.19) implies r¯n,λ1 ,λ2 = O(nBn /(log Bn )2 ). By orthogonality, √ √ n √ (λ) − Nn (λ) = O( nm), we have |rn,λ1 ,λ2 | ≤ |r n,λ1 ,λ2 | + O(n Bn m + 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), using Condition 3, we can show that lim sup 2(n Bn )−1 n→∞

n t−m−1

∑ ∑

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

t=1 s=1

Hence (ii) follows. ı jλ (iii) Recall that D0,λ 2 = ∑m j=−m E(X 0,m X j,m )e . From the proof of (i) and Condition 3, we see that for Bn−1 (log Bn )2 ≤ λ ≤ π − Bn−1 (log Bn )2 ,

r n (λ, λ) = O(n Bn /(log Bn )2 ) + D0,λ 42 n

Bn

∑

K 2 (s/Bn )

s=−Bn

= O(n Bn /(log Bn )2 ) + 4π 2 f 2 (λ)n Bn κ.

n

Hence (iii) holds.

LEMMA A.6. Recall (A.17) for uj (λ) and set E n = Bn − (log Bn )2 . Under the conditions of Theorem 3 or 4 or 5, we have P

| ∑kn uj (λ∗ )|2 max(log Bn )2 ≤i≤En 2j=1 2 i ∗ − 2 log(Bn ) + log(π log Bn ) ≤ x 4π n Bn f (λi )κ

→ e−e

−x/2

.


1243

k

1 n K 2 (t)dt = 1. Let Vn = ∑ j=1 Vj , where Proof. For convenience we assume κ = −1

u j (λi∗1 ), . . . , f −1 (λi∗d ) u j (λi∗d )), Vj = ( f −1 (λi∗1 )

1 ≤ j ≤ kn ,

(log Bn )2 ≤ i 1 < · · · < i d ≤ En . By Fact 2.2 in Einmahl and Mason (1997), there exist independent centered normal random vectors N1 , . . . , Nkn with Cov(Vj ) = Cov(N j ), 1 ≤ j ≤ kn , such that

3 P (|Vn − Nn | ≥ τn ) = O e−(log Bn ) , k

n where Nn = ∑ j=1 N j . For z = (z 1 , . . . , z d ) define |z|d = min{z i : 1 ≤ i ≤ d}. Then

3 (A.20) P |Vn |d ≥ yn n Bn ≤ P |Nn |d ≥ yn n Bn − τn + O e−(log Bn ) .

Let Id be a d × d identity matrix. We claim that |Cov(Nn ) − 4π 2 n Bn Id | = O(n Bn /(log Bn )2 ).

(A.21)

To prove (A.21), we first show that for |i k − il | ≥ (log Bn )2 /Bn , kn kn ∗ ∗ E ∑ uj (λi k ) ∑ uj (λil ) = O(n Bn /(log Bn )2 ). j=1 j=1

(A.22)

n ( u j (λi∗k ) − u j (λi∗k ))|2 = O(n −c ) for any c > 0. In fact, by (A.18), we have maxi k E| ∑ j=1

k

k +1

k +1

n n v j (λ)2 = ∑ j=1 v j (λ)2 . We now Recall (A.12) for v j (λ). By independence, ∑ j=1 t−4m ∗ estimate v j (λ)2 . Let Y´t,m (λ) = X t,m ∑t−1 s=t−4m+1 an,t−s X s,m , Yt,m (λ) = X t,m ∑s=1 an,t−s X s,m . As in the proof of (A.2), routine calculations yield that, for 1 ≤ j ≤ kn ,

2

2

∗ (λ) + 2 max Yt,m (Y´t,m (λ) − EY´t,m (λ)) max v j (λ)2 ≤ 2 max λ λ λ t∈I j t∈I j

∑

= O(m Bn2 ) + m

∑

∑ Y´t,m (λ)2 = O(m Bn2 )

t∈I j

2+2β

and maxλ v kn +1 (λ)2 = O m Bn

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

kn +1 v j (λ)|2 = O(n Bn1−ε ) for some ε > 0. So we ciently small numbers, imply maxλ E| ∑ j=1

have

n uj (λi∗k ) − g n,m (λi∗k )|2 = O(n Bn maxi k E| ∑ j=1

k

1−ε/2

).

We next prove that max E|gn,m (λ) − g n,m (λ)|2 = O(n Bn /(log Bn )2 ). λ

(A.23)

1244


Let It = ∑t−m s=1 an,t−s (X s,m − X s,m ). As in the proof of (A.2), 2 n n max ∑ X t,m It ≤ Cm max ∑ X t,m 2 It 2 λ t=1 λ t=1 = O(m 2 n Bn )E(X s,m − X s,m )2 = O(m 2 (n Bn )1−2α ) and, for Jt = ∑t−1 s=t−m+1 an,t−s (X s,m − X s,m ), 2 n max ∑ X t,m Jt ≤ Cm 3 nX s,m − X s,m 2 = O(m 3 n(n Bn )−2α ). λ t=1 Hence maxλ ∑nt=1 X t,m (It + Jt ) = O(τn ). Similarly, we can also have ∑nt=1 (X t,m − X t,m ) ∑t−1 s=1 an,t−s X s,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 (λi∗k ), gn (λi∗l ))| = O(τn2 ), which follows from Lemma A.5(i) immediately. Similarly, from Lemma A.5(iii) we have 2 kn ∗ 2 2 ∗ E ∑ uj (λ ) − 4π n Bn f (λ ) = O(τ 2 ). n ik ii j=1 This together with (A.22) yields (A.21), and hence

n Bn /(log Bn )2 . Cov1/2 (Nn ) − 2π n Bn Id = O

(A.24)

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

2 P Cov1/2 (Nn ) − 2π n Bn Id ||N ≥ τn = O(e−(log Bn ) /4 ), which, together with (A.20), yields that

2 P |Vn |d ≥ yn n Bn ≤ P 2π n Bn |N |d ≥ yn n Bn − 2τn + O e−(log Bn ) /4 = (1 + o(1))

√

8π yn−1 exp

y2 − n2 8π

d .

(A.25)

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

√ yn2 −1 8π yn exp − 2 P |Vn |d ≥ yn n Bn = (1 + o(1)) 8π

(A.26)

uniformly on {(λi∗ , . . . , λi∗d ), (log Bn )2 ≤ i 1 < · · · < i d ≤ En , |λi∗j − λi∗k | ≥ (log Bn )2 /Bn , 1 k = j}.


1245

By Lemma A.5(ii) and Lemma 2 in Berman (1962), similarly to (A.25) we have kn ∗ ∗ P ∑ uj (λi k ) ≥ yn n Bn f (λi k ), k = 1, . . . , d j=1 ≤C

√

8π yn−1 exp

y2 − n2 8π

d−2

× yn−2 exp

y2 − n2 (1 + δ) 8π

(A.27)

for some δ > 0, uniformly on {λi∗ − λi∗ ≥ Bn−1 , λi∗k − λi∗ ≥ Bn−1 (log Bn )2 ; k = 3, . . . , d}. 2 1 k−1 The details of the derivation are omitted. Let tn = 2 log Bn − log(π log Bn ) + x. Define ⎫ ⎧ ⎬ ⎨ | ∑kn uj (λ∗ )|2 & i j=1 Ai = and A = Ai . ≥ tn ∗ 2 2 ⎭ ⎩ 4π n Bn f (λi ) 2 (log Bn ) ≤i≤En

By Bonferroni’s inequality, we have for every fixed k that 2k

2k−1

t=1

t=1

∑ (−1)t−1 Pt ≤ P(A) ≤ ∑

(−1)t−1 Pt ,

where Pt = ∑(log Bn )2 ≤i 1