THE INTEGRATED PERIODOGRAM FOR STABLE ... - Project Euclid

The Annals of Statistics 1996, Vol. 24, No. 5, 1855–1879

THE INTEGRATED PERIODOGRAM FOR STABLE PROCESSES 1 ¨ By Claudia Kluppelberg and Thomas Mikosch

University of Mainz and University of Groningen We study the asymptotic behavior of the integrated periodogram for α-stable linear processes. For α ∈ 1; 2 we prove a functional limit theorem for the integrated periodogram. The limit is an α-stable analogue to the Brownian bridge. We apply our results to investigate some specific goodness-of-fit tests for heavy-tailed linear processes.

1. Introduction. In this paper we consider the linear process ∞ X Xt = (1.1) ψj Zt−j ; t ∈ Z; j=−∞

where Zt t∈Z is a noise sequence of iid symmetric α-stable r.v.’s for α ∈ 0; 2. This implies in particular that Zt and Xt have infinite variance. In two preced¨ ing papers [Kluppelberg and Mikosch (1993, 1994)] we studied the asymptotic behavior of the periodogram n X −iλt 2 −2/α λ ∈ −π; π: Xt ; In; X λ = n e t=1

The results obtained there indicate that the self-normalized periodogram e λ = In; X λ ; λ ∈ −π; π; I n; X 2 γn; X

with

2 −2/α γn; X =n

n X

X2t

t=1

behaves very much like the periodogram for finite variance linear processes. e λ converge in More precisely, for all α ∈ 0; 2, smoothed versions of I n; X probability to ψλ2 /ψ2 , where 2 ∞ X 2 −ijλ (1.2) λ ∈ −π; π; ψλ = ψj e ; j=−∞

denotes the power transfer function of the linear filter ψj j∈Z and ψ2 =

∞ X

ψ2j :

j=−∞

Received December 1993; revised January 1995. supported in part by New Zealand FRST Grant 93-VIC-36-039. AMS 1991 subject classifications. Primary 62M15, 62G07; secondary 60F17, 62M10. Key words and phrases. Linear process, moving average process, stable process, frequency domain, integrated periodogram, functional limit theorem, quadratic form, goodness-of-fit test. 1 Research

1855

¨ C. KLUPPELBERG AND T. MIKOSCH

1856

In the finite variance case ψ·2 is, up to a constant multiple, the spectral density of the linear process. In the case α < 2 this function can naturally not be interpreted as a spectral density of Xt . In the present paper we continue our investigation of the spectral analysis of α-stable processes. Our results can be understood as a study of classical (i.e., finite variance) quantities in spectral analysis when some of the innovations Zt assume very large values. In this sense, the theory given below provides some recommendations on how classical estimators and test statistics have to be modified when large Zt occur. As appropriate techniques we propose random normalization (we call it self-normalization) and random centering. In the sequel we consider modifications of the integrated periodogram Zx (1.3) In; X λfλ dλ; x ∈ −π; π; −π

e and the corresponding self-normalized version with In; X replaced by I n; X for smooth weight functions f. In the finite variance case, the integrated periodogram serves as an estimate of the spectral distribution function. In analogy to empirical process theory, this suggests building up goodness-of-fit tests of Kolmogorov–Smirnov or Cramér–von Mises type which are based on the integrated periodogram. This idea has been utilized for a long time, for example, by Bartlett, Grenander and Rosenblatt [see Priestley (1981) and Dzhaparidze (1986) and the recent account by Anderson (1993)]. It has been observed by several authors [cf. Anderson (1993) and the references therein] that the asymptotic theory for the standard goodness-of-fit test statistics is actually a consequence of a functional central limit theorem for the integrated periodogram. Thus the asymptotic distribution of the test statistics follows from such a limit theorem in the same way that the asymptotic distribution of Kolmogorov–Smirnov or Cramér–von Mises statistics follows from Donsker’s empirical central limit theorem. This idea also applies in the infinite variance case. To give an example, we shall prove that, for a positive constant Cα and α ∈ 1; 2, 1/α Z · ∞ X n sin·t (1.4) In; X λ − ψλ2 Tn dλ →d S· ≡ 2 Zt Cα log n t −π t=1 in C−π; π, the space of continuous functions on −π; π equipped with the uniform topology. Here 1 Z π In; X λ Tn = dλ: 2π −π ψλ2

For α = 2 the limit process in (1.4) is a Brownian bridge [cf. Hida (1980)], and this is analogous to the result of Anderson (1993) who obtained for the self-normalized integrated periodogram without random centering a Brownian bridge plus an additional Gaussian term. Random centering with Tn takes care of this additional term and makes the limit theory for the integrated periodogram very much like empirical process theory. We mention that a similar argument applies to the finite variance case.

INTEGRATED PERIODOGRAM FOR STABLE PROCESSES

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From (1.4) we can, for example, immediately derive the limit distribution of the Grenander–Rosenblatt statistic for α-stable Xt : Z 1/α x n 2 In; X λ − ψλ Tn dλ →d sup Sx: sup Cα log n −π≤x≤π −π≤x≤π −π

Tests for α-stable noise or ARMA processes follow easily. This shows the power of the functional central limit theorem (1.4) which reduces the problem of approximating the distribution of the Grenander–Rosenblatt statistic to a study of the properties of the limit process S·. Our paper is organized as follows. In Section 2 we introduce some general notation and assumptions. The main theoretical results, in particular the basic functional central limit theorem for the integrated periodogram (Theorem 3.2), are formulated in Section 3. In Section 4 we discuss some goodness-of-fit tests for α-stable Xt and illustrate the efficiency of the asymptotic theory by some computer simulations. In particular, we provide tables for the quantiles of the limit distribution of several test statistics. These tables demonstrate the dramatic contrast between the finite and the infinite variance cases. They are followed by Section 5 which contains some auxiliary results for the proofs in Section 6.

2. Assumptions and notation. We consider the moving average process Xt t∈Z defined by (1.1), where Zt t∈Z is a noise sequence of iid symmetric α-stable r.v.’s for α ∈ 0; 2. This means that the characteristic function of Z1 is given by α EeitZ1 = e−σt ; t ∈ R;

with a scaling factor σ > 0. For the definition and properties of α-stable r.v.’s, we refer to Feller (1971), Bingham, Goldie and Teugels (1987) or Petrov (1975). We mention that the restriction to symmetric stable r.v.’s is only for ease of presentation. Results can also be derived in the domain of attraction of stable r.v.’s or simply under moment conditions although the proofs then become much ¨ more technical [see, e.g., Mikosch, Gadrich, Kluppelberg and Adler (1995)]. In order to guarantee the a.s. absolute convergence of (1.1), we introduce the assumption ∞ X j ψj δ < ∞ (2.1) j=−∞

for some δ < min1; α. Condition (2.1) is obviously satisfied for every causal invertible ARMA process. Throughout we will also suppose that ψλ2 is everywhere positive. The following notation will be used throughout the paper: for any sequence of r.v.’s At t∈Z , n X 2 −2/α γn; A2t ; A = n t=1

2 n X −iλt −2/α In; A λ = n ; At e t=1

λ ∈ −π; π ;


1858

e = n−1/α A /γ A t n; A = At t

n . X

A2s

s=1

1/2

;

n 2 X −iλt 2 e e In; A λ = In; A λ/γn; A = At e ;

λ ∈ −π; π:

t=1

We introduce

yn = Cα n log n1/α ; where    

1 − ασ ; 202 − α cosπα/2 Cα =   σ; π

and we set

if α 6= 1; if α = 1;

n−k

γn; A k = yn−1 In particular, with xn =

X

k ≥ 1:

At At+k ;

t=1

n Cα log n

1/α

;

we obtain that 2 xn In; A λ − γn; A = 2

n−1 X t=1

γn; A t cosλt;

n ≥ 1:

The following proposition is Theorem 3.3 in Davis and Resnick (1986). It is a key result for the present paper. The normalizing sequence yn n∈N is a consequence of Corollary 2.1 in Rosinski and Woyczynski (1987) and of formula (3.4) in Davis and Resnick (1986). Proposition 2.1. Let Zt t∈Z be a noise sequence satisfying the assumptions above. Let Y0 be an α/2-stable positive r.v. which is independent of Zt t∈Z and has Laplace transform E exp−rY0 = exp−σKα rα/2 ; where Kα = ENα/2 for an N0; 2-r.v. Then 2 γn; Z ; γn; Z k; k = 1; : : : ; m →d Y0 ; Zk ; k = 1; : : : ; m;

m ≥ 1:

¨ For a derivation of the Laplace transform of Y0 , we refer to Kluppelberg and Mikosch (1994), Proposition 3.2.


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3. Main results. We start with a consistency result for both the integrated periodogram and the self-normalized integrated periodogram. In the sequel C−π; π denotes the space of the continuous functions on −π; π equipped with the sup-norm. Theorem 3.1. Let Xt t∈Z be defined by (1.1) for some α ∈ 0; 2 and suppose that the coefficients ψj j∈Z satisfy condition (2.1). Let f be a nonnegative 2π-periodic continuous function such that the Fourier coefficients of f·ψ·2 are absolutely summable. Then Z · Z · 2 2 2 In; X λfλ dλ →d Y0 ψ ; 1; (3.1) γn; X ; Tn ; ψλ fλ dλ −π

−π

in R1 × R1 × C−π; π, where the r.v. Y0 is defined in Proposition 2.1. In particular, Z 2 x e λ − ψλ fλ dλ → 0: (3.2) sup I n; X P 2 ψ −π≤x≤π −π

Remark 3.1. By a Cramér–Wold argument the statement of Theorem 3.1 also holds for general functions f provided both the positive part and the negative part of f satisfy the conditions of Theorem 3.1. 2 2 Remark 3.2. Note that, since γn; X /ψ − Tn →P 0, one can work with Tn 2 instead of γn; X in results of type (3.2). This is no longer true if one is interested in the rate of convergence in (3.2) (see Theorem 3.2 below). Indeed, for a finite 2 2 moving average process, it is not difficult to see that xn γn; X /ψ − Tn does not converge weakly to 0.

Remark 3.3. Note that Theorem 3.1 is valid for all α ∈ 0; 2. In the classical case where Z1 has a second moment, the result remains formally true with Y0 replaced by 2σ 2 . This is a consequence of the law of large numbers. e , we are not able to distinguish Thus, if we use the self-normalized version I n; X between an infinite variance α-stable process Xt t∈Z and a process with finite second moment. This can be interpreted as a robustness property of the self-normalized periodogram which has intuitively been known for a long time [see Priestley (1981)]. On the other hand, (3.1) and (3.2) show the profound difference in the limit behavior of the periodogram and the self-normalized version. The contrast between the infinite and the finite variance case again manifests in the rates of convergence in Theorem 3.2 below. Example 3.1.

For f· = ψ·−2 we obtain from Theorem 3.1 that Z e x In; X λ 1 sup dλ →P 0: − 2 2 ψλ ψ −π≤x≤π −π

This result holds for any causal invertible ARMA process.


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Example 3.2. Therefore,

The function f ≡ 1 satisfies the assumptions of Theorem 3.1. Z 2 x ψλ e sup dλ →P 0: In; X λ − 2 ψ −π≤x≤π −π

Theorem 3.2. Let Xt t∈Z be defined by (1.1) with coefficients ψj j∈Z satisfying (2.1) and suppose that α ∈ 1; 2. Furthermore, assume that f is defined on −π; π such that g· = f·ψ·2 is continuously differentiable. Then Z · 2 2 γn; ; T I λ − ψλ T fλ dλ ; x n n; X n n X →d

(3.3)

=d 1

1

−π

2

Y0 ψ ; Y0 ; 2

∞ X

t=1 2

Zt

Z

−π

Y0 ψ ; Y0 ; g·S· −

in R × R × C−π; π, where S· = 2

(3.4)

·

Z

gλ costλ dλ ·

−π

g λSλ dλ 0

∞ X sin·t Zt t t=1

and Y0 is independent of Zt t∈Z with the same distribution as in Proposition 2.1. Remark 3.4. Notice that the representation (3.4) is analogous to the Lévy– Cieselski or Paley–Wiener representation of a Brownian bridge [e.g., Hida (1980)]. Indeed, if Zt t∈Z is iid Gaussian white noise, then (3.4) represents a Brownian bridge. Remark 3.5. The series S is a random Fourier series with α-stable coefficients and as such it is a harmonizable stable process with discrete spectral measure. As a limit process in C−π; π, it has continuous sample paths. For its tails we conclude that P sup Sx > t = Ot−α ; t → ∞: −π≤x≤π

Figure 1 shows simulated sample paths of the process Sx0≤x≤π for different values of α. Some of them look surprisingly regular, at a first sight almost deterministic like slightly perturbed sine curves. For comparison the first picture shows a simulated Brownian bridge, that is, a path of S for α = 2. The sample paths for α < 2 behave completely differently. According to our limited experience from simulations, “deterministic” behavior appears more often with decreasing α. The sine shape can be well explained: since the distribution tail of the r.v. Z1 is very heavy, some of the Zt will be huge compared with the others. Thus the contribution of the sine function Zt sin·t/t to the series S is larger the larger the value of Zt and the smaller t. We consider some particular cases of Theorem 3.2. First we choose f ≡ 1.

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-2

S(x) for alpha=1.8 0 2 4 6

Brownian bridge -0.5 0.0 0.5 1.0


0.5

1.0 1.5 2.0 Frequency x

2.5

3.0

0.0

0.5


2.5

3.0

0.0

0.5


2.5

3.0

0.0

0.5


2.5

3.0

S(x) for alpha=1.5 -10 -8 -6 -4 -2 0

S(x) for alpha=1.2 -20 0 20 40

0.0

Fig. 1. Sample paths of the process S on 0; π with α-stable noise.

Corollary 3.3. Under the assumptions of Theorem 3.2, we obtain that Z · Z · ∞ X xn In; X λ − ψλ2 Tn dλ →d 2 Zt ψλ2 costλ dλ; −π

xn

Z

·

−π

In; X λ − ψλ2 Tn

t=1 ∞ X

−π

Zt Z · dλ →d 2 ψλ2 costλ dλ: Y −π t=1 0

Now we choose f· = ψ·−2 . Corollary 3.4. Under the assumptions of Theorem 3.2, we obtain that Z · I n; X λ − T xn n dλ →d S·; ψλ2 −π Z · I S· n; X λ xn : − 1 dλ →d 2 Y0 −π Tn ψλ Remark 3.6. Corollary 3.4 is analogous to the finite variance case where the limit process is a Brownian bridge [see Grenander and Rosenblatt (1957), Priestley (1981) and Dzhaparidze (1986)].


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In the case α ∈ 0; 1 we cannot expect a result which is similar to Theorem 3.2. For example, note that ∞ X sinxt α 1/α Sx =d 2Z1 ; t t=1

where the series on the right-hand side diverges except for x = 0; π; −π. Nevertheless, in some cases it is possible to determine the limit law of the integrated periodogram for a fixed frequency. This follows from the next result. Proposition 3.5. Let Xt t∈Z be a linear process as defined in (1.1) with coefficients ψj j∈Z satisfying (2.1) and suppose that α ∈ 0; 2. Furthermore, assume that f is defined on −π; π such that g· = f·ψ·2 is continuous and µ ∞ Z x X (3.5) −π gλ costλdλ < ∞ t=1

for some x ∈ −π; π and some 0 < µ < α. Then Zx 2 2 γn; ; T ; x I λ − ψλ T fλ dλ n n n; X n X →d =d

−π

Y0 ψ2 ; Y0 ; 2

∞ X

Zt

t=1 2

Y0 ψ ; Y0 ; 2Z1

Z

x

−π

gλ costλ dλ

α 1/α ∞ Z x X gλ costλ dλ ; −π

t=1

with Y0 independent of Zt t∈Z and with the same distribution as in Proposition 2.1. 4. Goodness-of-fit tests for a-stable processes. In this section we review some specific goodness-of-fit tests and study their asymptotic behavior under the assumption of heavy-tailedness. Note that all test statistics are continuous functionals of the integrated periodogram. Hence their weak limits are immediate consequences of Theorem 3.2 (see also Corollaries 3.3 and 3.4). The following results hold under the assumptions of Theorem 3.2 for the linear process (1.1). They are modifications of tests which are well studied in the finite variance case [e.g., Priestley (1981), subsection 6.2.6]. 1. Grenander and Rosenblatt’s test for the nonnormalized integrated spectrum: Z x 2 xn sup In; X λ − ψλ Tn dλ −π≤x≤π −π ∞ Zx X 2 →d sup 2 ψλ cosλt dλ : Zt −π≤x≤π

t=1

−π


Test for α-stable noise: Z x In; X λ − Tn dλ →d xn sup −π≤x≤π −π

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sup Sx:

−π≤x≤π

2. Bartlett’s test for the self-normalized integrated spectrum: Z x ψλ2 Tn e In; X λ − xn sup dλ 2 γn; X −π≤x≤π −π ∞ Zx X 1 2 : 2 ψλ cosλt dλ Z sup →d t 2 Y0 ψ −π≤x≤π t=1 −π Test for α-stable noise: Z x Tn e xn sup In; X λ − 2 dλ →d γn; X −π≤x≤π −π

sup −π≤x≤π

Sx : ψ2 Y0

3. Bartlett’s Tp test:

Z x In; X λ xn sup − 1 dλ →d 2T ψλ −π≤x≤π −π n

2

sup −π≤x≤π

Sx : Y0

4. ω -statistic or Cramér–von Mises test: 2 Zπ Z π Z x I ∞ X Z2t n; X λ 2 dλ S x dx = 4π dx → − T : x2n d n ψλ2 t2 −π −π −π t=1

5. ω2 -test or Cramér–von Mises test with normalization Tn for the integrated spectrum: 2 Z π Z x I ∞ e λ Z2t 4π X 1 Zπ 2 n; X 2 S x dx = xn : − 1 dλ dx → d 2 −π −π ψλ Tn Y02 −π Y02 t=1 t2

The above results can immediately be compared with the finite variance case. In the classical situation the normalizing constants xn are of the order √ n, and S represents a Brownian bridge [see Hida (1980), Priestley (1981) and Dzhaparidze (1986)]. Formally, the limits in the cases α ∈ 1; 2 and α = 2 are the same if we replace Y0 by 2σ 2 for α = 2. The rate of convergence in the α-stable case compares favorably with the finite variance case. This seems to be the rule for the limit theory of α-stable time series, in the time domain, in the frequency domain, but also in estima¨ tion theory for linear processes. We refer to Mikosch, Gadrich, Kluppelberg and Adler (1995) for a discussion of this phenomenon and more references. However, the limit distributions of the above statistics are much less common than in the finite variance case. Their study is by trivial. P no means 2 2 The distribution of the quadratic functional ∞ Z /t is characterized t=1 t by the relation α ∞ ∞ X X Z2t Nt 2/α −1 = Y Kα : t t2 d 0 t=1 t=1

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Here Y0 is α/2-stable as defined in Proposition 2.1 and Nt t∈Z are iid N0; 2Gaussian r.v.’s independent of Y0 . Indeed, for Nt t∈Z independent of Zt t∈Z , we have, for positive λ, ∞ ∞ X X Zt Nt Z2t 1/2 = E exp iλ E exp −λ t2 t t=1 t=1 ∞ X Nt α 1/α 1/2 = E exp iλ Z1 t t=1 α ∞ X N = E exp −σλα/2 Kα Kα−1 t t t=1 ∞ α 2/α X Nt = E exp −λY0 K−1 : α t t=1

In particular, let Y0 and Y1 be iid and independent of Nt t∈Z . Then we conclude that α 2 Z π Z x I ∞ e λ X Nt 2/α 1 4πY0 n; X −1 2 Kα : − 2 dλ dx →d xn t ψλ2 ψ −π −π ψ4 Y21 t=1

We conclude this section with a short simulation study in order to show the efficiency of the asymptotic theory given above. The study was carried out in S-Plus on a SUN workstation and we chose the parameters (sample size, number of discretization points and terms of the series) within the limits of available memory and computer time. The theoretical evaluation of the distributions of continuous functionals of S· seems very difficult and calls for further research. We restrict ourselves to some simulation studies in order to show that the proposed methods already work for medium sample sizes n between 200 and 400. For obvious symmetry reasons, we only consider the process S· and its functionals on the interval 0; π. The dramatic contrast between the cases α < 2 and the Brownian bridge, visualized in Figure 1, is also well illustrated by the following tables. In Table 1 we give the quantiles of the limit distribution Aα = sup Sx 0≤x≤π

for statistics of Grenander–Rosenblatt type with α-stable innovations Zt . For reasons of comparability, we include also the case α = 2 which corresponds to a Brownian bridge. The quantiles for the Brownian bridge were calculated from the corresponding ones of the absolute supremum functional of the standard Brownian bridge on 0; 1 provided in Smirnov (1948); see Shorack and Wellner (1986), page 143. The quantiles for α = 1:2 and α = 1:8 are the empirical ones from 800 independent simulations of Aα with 700 terms of the series S. In Table 2 we give the quantiles of the limit distribution Zπ Bα = S2 x dx 0


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Table 1 Quantiles of the absolute supremum functional of S on 0; π Quantile

A2

A1:8

A1:2

0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50 0.55 0.60 0.65 0.70 0.75 0.80 0.85 0.90 0.91 0.92 0.93 0.94 0.95 0.96 0.97 0.98 0.99

2.31 2.53 2.71 2.89 3.02 3.15 3.29 3.42 3.55 3.69 3.82 4.00 4.13 4.31 4.53 4.75 5.06 5.42 5.55 5.64 5.78 5.91 6.04 6.22 6.44 6.75 7.20

3.43 3.82 4.04 4.26 4.50 4.73 4.97 5.19 5.40 5.70 5.96 6.21 6.51 6.74 7.17 7.52 8.08 8.79 9.24 9.56 9.84 10.11 10.68 11.38 12.39 14.07 17.69

6.99 7.82 8.61 9.36 10.00 10.80 11.51 12.28 13.17 14.23 15.21 16.25 17.60 19.09 20.90 23.87 30.00 39.44 44.22 48.16 51.53 58.77 61.98 73.95 85.71 103.4 180.3

of statistics of ω2 -type with α-stable innovations Zt . The quantiles for the Brownian bridge were calculated from the corresponding ones of the ω2 -functional of the standard Brownian bridge on 0; 1 provided in Anderson and Darling (1952); see Shorack and Wellner (1986), page 147. The quantiles for α = 1:2 and α = 1:8 are the empirical ones from 50,000 independent simulations of Bα with 600 terms in the series representation. In a simulation study we convinced ourselves that the goodness-of-fit tests work well in the heavy-tailed case for medium sample sizes of 200 to 400. In a first experiment we considered the Grenander–Rosenblatt statistic for 1:8-stable noise modified for the interval 0; π; that is, Z x An = sup In; Z λ − Tn dλ : 0≤x≤π

−π

In Figure 2, for a realization Zt of 1:8-stable noise, the values An are plotted against n (thin line). For comparison, the values 10:68/xn (thick line) are also plotted. The value 10.68 stands for the 0.95 quantile of the distribution of A1:8 (see Table 1).


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Table 2 Quantiles of the ω2 -functional of S on 0; π Quantile

B2

B1:8

0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50 0.55 0.60 0.65 0.70 0.75 0.80 0.85 0.90 0.91 0.92 0.93 0.94 0.95 0.96 0.97 0.98 0.99

2.27 2.85 3.36 3.86 4.36 4.87 5.42 6.01 6.66 7.37 8.17 9.09 10.16 11.43 12.98 14.96 17.61 21.54 22.58 23.77 25.13 26.71 28.61 30.96 34.03 38.43 46.10

10.66 13.58 16.06 18.43 20.88 23.38 26.11 29.06 32.31 36.07 40.21 45.22 50.87 57.75 66.54 78.50 95.99 125.3 134.5 144.6 157.3 172.6 195.4 224.9 281.3 383.7 747.6

B1:2 34.94 46.97 58.34 70.05 83.15 98.24 116.1 135.5 159.6 190.6 228.5 278.3 350.9 447 602 852 1,334 2,605 3,056 3,694 4,520 5,873 7,799 11,109 17,955 36,548 110,210

In a second experiment we simulated realizations of an AR(1) process Xt = 0:5Xt−1 +Zt with 1:8-stable noise Zt and calculated the ω2 -statistic modified for the interval 0; π; that is, 2 Z π Z x I n; X λ Bn = dx: − T n dλ ψλ2 0 −π

In Figure 3 we plotted the values Bn (thin line) against 195.4/x2n (thick line) where 195.4 is the 0.95 quantile of B1:8 (see Table 2). 5. Auxiliary results. The following statement is part of Proposition 4.3 in Davis and Resnick (1986). Lemma 5.1.

Under (1.1) and (2.1), 2 2 2 γn; X − ψ γn; Z →P 0:

We will frequently make use of the following decomposition of the peri¨ odogram [see Kluppelberg and Mikosch (1993), Proposition 2.1].

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Grenander-Rosenblatt statistic / 0.95-quantile 0.2 0.4 0.6 0.8 1.0


0

200

400 600 Sample size n

800

1000

Fig. 2. The Grenander–Rosenblatt statistic An thin line and the 0:95 quantile of A1:8 /xn thick line.

Lemma 5.2.

Under (1.1) and (2.1),

In; X λ = ψλ2 In; Z λ + Rn λ;

−π < λ ≤ π;

where 2 Rn λ = ψλ Jn λ Yn −λ + ψ−λ Jn −λ Yn λ + Yn λ ; Jn λ = n−1/α

Yn λ = n−1/α n−j

Unj λ =

X

t=1−j

n X

Zt e−iλt ;

t=1 ∞ X

j=−∞

ψj e−iλj Unj λ;

Zt e−iλt −

n X

Zt e−iλt :

t=1

¨ From the proof of Lemma 9.2 in Mikosch, Gadrich, Kluppelberg and Adler (1995), we conclude the following result [which has been proved there for


0.0

Cramer-von-Mises statistic / 0.95-quantile 0.5 1.0 1.5

2.0

1868

0

200

400 600 Sample size n

800

1000

Fig. 3. The Cramér–von Mises statistic Bn thin line and the 0:95 quantile of B1:8 /x2n thick line.

ARMA processes, but a careful study of the proof shows that it remains valid for more general processes Xt t∈Z ]: Lemma 5.3. Under (1.1) and (2.1), the remainder Rn defined in Lemma 5.2 satisfies the relation Zπ xn Rn λ dλ →P 0: −π

From Lemmas 5.2 and 5.3 we conclude the following. Lemma 5.4.

Under (1.1) and (2.1), 2 xn Tn − γn; Z →P 0:

We also need the following elementary tool.

Lemma 5.5. Let G1 ; G2 be jointly Gaussian with zero mean and identical variances. Then G1 G2 =d 12 VarG1 N12 − N22 + 12 CovG1 ; G2 N12 + N22 ;

where N1 ; N2 are iid N0; 1-r.v.’s.


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Proof. The quadratic form G1 G2 = 12 G1 G2 + G2 G1 = GTAG; with A=

1 2

0 1

1 ; 0

G=

G1 ; G2

can be written as z1 N12 + z2 N22 for iid standard Gaussian N1 ; N2 and for the eigenvalues z1 ; z2 of the matrix 6A, where 6 is the covariance matrix of G. The eigenvalues of 6A are z1 = 12 VarG1 + CovG1 ; G2 and z2 = 12 CovG1 ; G2 − VarG1 : Hence z1 N12 + z2 N22 is distributed as 1 VarG1 N12 2

− N22 + 12 CovG1 ; G2 N12 + N22 :

2

6. Proofs. Throughout, c will denote a generic positive constant which may change from formula to formula or from line to line. Proof of Theorem 3.1. We define g2 · = f·ψ·2 . In view of Lemmas 5.1 to 5.4 we have, uniformly for x ∈ −π; π; Zx 2 γn; X ; Tn ; In; X λfλ dλ −π

Z 2 2 = 1 + oP 1 ψ2 γn; Z ; γn; Z ;

x

−π

In; Z λg2 λ dλ :

Here we also used the boundedness of f and ψλ2 . We define Z · Mn · = In; Z λg2 λ dλ −π Z · = ReJn λ2 + ImJn λ2 g2 λ dλ −π

and show the fidi-convergence (6.1)

Z 2 γn; ; M · → Y n d 0 1; Z

·

−π

g2 λ dλ :

We restrict ourselves to the case of two dimensions. Suppose −π < x < y ≤ π. Let Bi , i = 1; 2; 3; 4; be independent Brownian motions on −π; π, Nt t∈Z


1870

be iid N0; 1-r.v.’s and let Zt , Bi and Ni be independent. Then, for real ri , i = 1; 2; 3; 2 r1 r23 2 r22 E exp − Mn x − Mn y − γn; Z 2 2 2 2Zx r ReJn λ2 + ImJn λ2 g2 λ dλ = E exp − 1 2 −π r2 2 r2 Z y ReJn λ2 + ImJn λ2 g2 λ dλ − 3 γn; − 2 Z 2 −π 2 Z x Zx ImJn λgλ dB2 λ ReJn λgλ dB1 λ + = E exp ir1 −π −π Z y Zy ImJn λgλ dB4 λ ReJn λgλ dB3 λ + + ir2 −π

−π

Z n X −1/α = E exp in Zt r1 t=1

+ ir3 n

−1/α

x

−π

+ r1 + r2 + r2

cosλtgλ dB1 λ

Z

n X

Zt Nt

t=1

x

−π Zy

−π Zy −π

sinλtgλ dB2 λ cosλtgλ dB3 λ sinλtgλ dB4 λ + r3 Nt

Z Zx n x σ X r1 = E exp − sinλtgλ dB2 λ cosλtgλ dB λ + r 1 1 n t=1 −π −π Zy cosλtgλ dB3 λ + r2 −π α Zy + r2 sinλtgλ dB4 λ + r3 Nt −π n σ X α =x E exp − G : n t=1 t

The r.v.’s

Gt = r1

Z

x

−π

+ r2

Z

cosλtgλ dB1 λ + r1 y

−π

Z

x

−π

cosλtgλ dB3 λ + r2

Z

sinλtgλ dB2 λ y

−π

sinλtgλ dB4 λ + r3 Nt

are Gaussian with expectation 0 and covariances (δts is the Kronecker symbol) CovGt ; Gs Zx Z = r21 cosλt − sg2 λ dλ + r22 −π

y

−π

cosλt − sg2 λ dλ + r23 δts :

1871


We have for a standard Gaussian r.v. N1 that Zx Z g2 λ dλ + r22 EGt α = EN1 α r21

y

−π

−π

g2 λ dλ + r23

= EN1 α VarG1 α/2

α/2

independent of t. Applying Lemma 5.5, we conclude that Gt Gs is distributed as 1 VarG1 N12 2

In order to show that

− N22 + 12 CovGt ; Gs N12 + N22 :

n−1

n X

t=1

Gt α →P EG1 α

holds, it suffices to show that 2 n n X X var n−1 Gt α = E n−1 Gt α − EG1 α → 0: t=1

t=1

Hence it suffices to show that the following expression converges to 0: 1 X EGt Gs α − EG1 α 2 n2 t6=s α 1 1 X 1 2 2 2 2 = 2 E VarG1 N1 − N2 + CovGt ; Gs N1 + N2 n t6=s 2 2 − EN1 α 2 VarG1 α :

Notice that N12 −N22 has the same distribution as 2N1 N2 . Using the continuity of the lα -seminorm, it suffices to show that X ECovGt ; Gs N12 + N22 α → 0: n−2 t6=s

We have, with positive constants c, 1 X ECovGt ; Gs N12 + N22 α n2 t6=s 1 X 2 Z x cosλt − sg2 λ dλ E r1 = 2 n t6=s −π α Zy 2 2 2 2 + r2 cosλt − sg λ dλ N1 + N2 −π

α Z Zy x c X cosλtg2 λ dλ ≤ 2 cosλtg2 λ dλ + r22 n − t r21 n 1≤tε t −π≤x≤π

≤P

≤P

(6.4)

n>t>m

b X

p=a b X

≤P

+

t=2

εp > ε

p=a

+P

2p+1 −1 X sinxt γn; Z t sup > ε t −π≤x≤π p

b [

p=a

b X

2p+1 −1 X sinxt > ε sup γn; Z t p t −π≤x≤π p

t=2

εp > ε

p=a

b X

P

p=a

= I1 + I2 :

2p+1 −1 X sinxt > ε γn; Z t sup p t −π≤x≤π p t=2

Moreover, 2p+1 −1 X sinxt Jp = P sup γn; Z t > εp t 0≤x≤π p

=P ≤ =

t=2

max

max

2p+1 −1 X sinxt > ε t γ p n; Z t −2p p

sup

k=1;:::;2p j∈k−12p +1;:::;k2p x∈j−1π2−2p ; jπ2

X

k=1;:::;2p

X

P

max

t=2

2p+1 −1 X sinxt γn; Z t > εp t −2p p

sup

j∈k−12p +1;:::;k2p x∈j−1π2−2p ; jπ2

Jpk :

t=2

k=1;:::;2p

We have Jpk = P

max

sup

2p+1 −1 X γn; Z t −2p p

j∈k−12p +1;:::;k2p x∈j−1π2−2p ; jπ2

t=2

sinx − j − 1π2−2p t cosj − 1π2−2p t t cosx − j − 1π2−2p t sinj − 1π2−2p t > ε + p t ×


=P

max

j∈k−12p +1;:::;k2p 0≤x≤2−2p π

× ≤P

sup

2p+1 −1 X γn; Z t p t=2

−2p

sinxt cosj − 1π2 max

sup

j∈k−12p +1;:::;k2p 0≤x≤2−2p π

t + cosxt sinj − 1π2−2p t > εp t

2p+1 −1 X γn; Z t p t=2

×

+P

max

1875

sup

j∈k−12p +1;:::;k2p 0≤x≤2−2p π

= Jpk1 + Jpk2 :

sinxt cosj − 1π2−2p t > ε /2 p t

2p+1 −1 X γn; Z t p t=2

cosxt sinj − 1π2−2p t × > εp /2 t

We restrict ourselves to the estimation of Jpk2 . We write · instead of sup0≤x≤2−2p π · and Yj · =

2p+1 X−1 t=2p

γn; Z t

cos·t sinj − 1π2−2p t : t

We intend to apply a maximal inequality to Jpk2 = P max Yj > εp /2 : j∈k−12p +1;:::;k2p

Note that for j < j0 both taken from k − 12p + 1; : : : ; k2p and for positive constants c, PYj − Yj0 > εp /2 2p+1 −1

X j − j0 −2p cos·t 2 sin π2 t γ t =P n; Z

t 2 t=2p

j + j0 − 2 −2p

× cos π2 t > εp /2 2 2p+1 −1 0

X j − j cos·t −2p

> ε c : sin π2 t ≤ cP γ t p n; Z

t 2 t=2p

In the last step we used the contraction principle for random quadratic forms [see Kwapien and Woyczynski (1992), Proposition 6.3.2]. For x < y both taken from 0; 2−2p π we have in view of Theorem 3.1 in Rosinski and Woyczynski


1876

(1987) or Theorem 6.9.4 in Kwapien and Woyczynski (1992) that 2p+1 −1 X cosyt cosxt j − j0 −2p P γn; Z t − sin π2 t > εp c t t 2 t=2p µ 2p+1 X−1 cosyt cosxt j − j0 −2p −µ − sin π2 t ≤ cεp t t 2 t=2p −µ ≤ cεp x − yµ j − j0 2−2p µ 2pµ+1

for an arbitrary 1 < µ < α and positive constants c. An application of the maximal inequality Theorem 12.2 in Billingsley (1968) yields that −µ −2pµ PYj − Yj0 > εp /2 ≤ cεp 2 j − j0 2−2p µ 2pµ+1

= cj − j0 2−2p µ 2p1+µ−1+2/γ and, again applying the same inequality, we obtain Yj > εp /2 Jpk2 = P max j∈k−12p +1;:::;k2p

≤ c2−pµ 2p1+µ−1+2/γ = c2p1+2µ−1+1/γ :

[Note that Theorem 12.2 in Billingsley (1968) is proved for real-valued Yj but a careful study of the proof shows that it remains valid for Banach space valued r.v.’s.] Hence we have X Jpk ≤ c22p1+µ−1+1/γ : Jp ≤ k=1;:::;2p

Now choose γ such that κ = −1 + µ−1 + 1/γ > 0: From (6.4) we conclude that for every ε > 0 and a sufficiently large lim sup I = lim sup I2 ≤ c n→∞

n→∞

∞ X

p=a

2−2pκ ≤ c2−2aκ :

This proves (6.3) and implies (3.3) for g ≡ 1. For general g the relation (3.3) follows by integration by parts and by the continuous mapping theorem. 2 The proof of Proposition 3.5 is essentially based on the proof of Lemma 9.3 ¨ in Mikosch, Gadrich, Kluppelberg and Adler (1995); it is reformulated as a result for quadratic forms in iid α-stable r.v.’s. Lemma 6.1. (6.5)

Let ft be real numbers such that ∞ X

t=−∞

ft µ < ∞

1877


for some µ < α. If f0 = 0, then ∞ X X 2 −1 γn; ; y → f Z Z f + f Z Y ; d t−s t s t −t t 0 n Z 1≤t;s≤n

t=1

=d If f0 6= 0, then

X

n−2/α

1≤t; s≤n

Y0 ; Z1

∞ X

t=1

ft + f−t α

1/α :

ft−s Zt Zs →d f0 Y0 :

Proof of Proposition 3.5. We have by Lemmas 5.1 to 5.4 that Zx 2 2 γn; X ; Tn ; xn In; X λ − ψλ Tn fλ dλ −π

2 2 = ψ2 γn; Z 1 + oP 1; γn; Z 1 + oP 1;

xn

Z

x

−π

In; Z λ −

2 γn; Z gλ dλ

+ oP 1

2 2 = ψ2 γn; Z 1 + oP 1; γn; Z 1 + oP 1; 2xn

n−1 X t=1

γn; Z t

Z

x

−π

cosλtgλ dλ + oP 1 :

Now we write ft =

Z

x

−π

cosλtgλ dλ

and apply Lemma 6.1 which concludes the proof. 2 As Proposition 3.5 shows, the statement of Theorem 3.2 can remain valid for certain frequencies, but not uniformly on −π; π: We demonstrate this with the process Z · I n; X λ xn (6.6) − T n cos λ dλ ψλ2 −π for a positive power transfer function ψ·2 satisfying (2.1). By (6.2) the problem of weak convergence of (6.6) can be reduced to the question of weak convergence of Z · 2 Dn · = xn In; Z λ − γn; Z cos λ dλ: −π


1878

Given that Dn converges weakly in C−π; π to a limit process of the form 2

∞ X

t=1

Zt

Z

·

−π

cos λ cosλt dλ =x

∞ X

t=1

Zt ft ·;

a necessary condition for its existence is that X ft xα < ∞ t

for all x ∈ −π; π. This series converges for x = −π; 0; π/2; −π/2; π but it does not for any other rational or irrational multiple of π. Hence for the centering sequence Tn weak convergence does not hold in the uniform topology for α ∈ 0; 1. Acknowledgments. The first named author would like to thank her colleagues at the Institute of Statistics and Operations Research for their kind hospitality during her stay at Victoria University, Wellington, in October/ November 1993. Both authors would like to thank the referee whose comments led to an improvement of the presentation of the results.

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Shorack, G. R. and Wellner, J. A. (1986). Empirical Processes with Applications to Statistics. Wiley, New York. Smirnov, N. V. (1948). Table for estimating the goodness of fit of empirical distributions. Ann. Math. Statist. 19 279–281. Department of Mathematics Johannes Gutenberg University D-55099 Mainz Germany

Faculty of Mathematics and Physical Sciences University of Groningen P.O. Box 800 Groningen The Netherlands