2 Weak convergence of the empirical spectral process

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EMPIRICAL SPECTRAL PROCESSES AND THEIR APPLICATIONS TO STATIONARY POINT PROCESSES

By Michael Eichler Universitat Heidelberg

Abstract

We consider empirical spectral processes indexed by classes of functions for the case of stationary point processes. Conditions for the measurability and equicontinuity of these processes and a weak convergence result are estab- lished. The results can be applied to the spectral analysis of point processes.

In particular, we discuss the application to parametric and nonparametric spectral density estimation.

1 Introduction

In the context of spectral analysis of time series, Dahlhaus (1988) introduced empirical processes where the spectral distribution function of a stationary process takes the part of the probability distribution. The asymptotic theory of these empirical spectral processes provides a method for proving limit theorems for statistics which depend on the spectral distribution.

In this paper, we are interested in empirical spectral processes derived from stationary point processes. Here a point process on ^Ris dened as a random counting measureN where N(A) denotes the number of point events occuring in some Borel set A ^Rcf. Daley and Vere-Jones (1988)]. In a fundamental paper by Brillinger (1972), it was shown that the spectral analysis of such processes based on nite Fourier transforms leads to similar results as in time series analysis. As an important dierence to the case of time series, we note that the cumulant spectra of point processes are functions on ^R which do not vanish for high frequencies and thus are not ^L²-integrable.

Consider a stationary point process N on ^R. If N satises certain mixing conditions, the spectral densityf² of N exists and is given by

f²() =^Z

R

exp(^;iu)dC²⁰(u) ²^R: (1.1) HereC²⁰ denotes the reduced cumulant measure of second order, which is dened by the equation

cum^fN(A¹)N(A²)^g=^Z_A

1 Z

A²dC²⁰(t¹^;t²)dt² (1.2)

AMS 1991 subject classications. Primary 60F05 secondary 62M15, 60G55.

Key words and phrases. Point processes, empirical spectral process, functional central limit theorem, spectral density estimation.

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for all A¹A² ² ^B. Now many interesting functionals in spectral analysis can be written in the form

A() =^Z

R

()f²()d

with parameter ² ^R^p. Examples we have in mind are the spectral distribution function F²() = ^R⁰f²()d, the covariance density function q²(u) =

R

Rexp(iu)(f²()^;p=2 )d and the variance time curve V (t) = var^fN((0t])^ge.g.

Brillinger (1975)].

If the process has been observed on the interval 0T], the spectral density can be estimated by the periodogram

I⁽^T⁾() =^f2 H²⁽^T⁾(0)^g^;1d⁽^T⁾()d⁽^T⁾(^;) where

d⁽^T⁾() =^Z

R

h⁽^T⁾(t)exp(^;it)dN(t)^;p^⁽^T⁾dt]

is the nite Fourier transform of the point process, h⁽^T⁾(t) = h(t=T) is a data taper with Fourier transforms

Hk⁽^T⁾() =^Z

R

h⁽^T⁾(t)^kexp(^;it)dt and p^⁽^T⁾ =^fH¹⁽^T⁾(0)^g^;1^Z

R

h⁽^T⁾(t)dN(t)

is an estimate for the mean intensityp of the process. The taper function h :^R^!^R is of bounded variation, vanishes outside the interval 01] and should be smooth with h(0) = h(1) = 0. However, our results also include the classical case where h(t) = 1⁰^1](t). We further dene

Hk =^Z

R

h(t)^kdt:

Substituting the periodogramI⁽^T⁾ for the spectral density, we obtain as an estimate for A()

A⁽^T⁾() =^Z

R

()I⁽^T⁾()d:

For nitely many such quadratic statistics have been studied e.g. by Brillinger (1972, 1978) and Tuan (1981). In these papers, the asymptotic normality of the estimateA⁽^T⁾() has been derived for the nontapered case.

The present paper deals with the case where the parameter space consists of innitely many parameters. More generally, we establish a functional central limit theorem for the empirical spectral process

E_T⁽^w⁾(g) =^pT^Z

R

g()I⁽^T⁾()^;f²()]w()d 2

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where g : ^R ^! ^C is from a suitable class of functions. The weight function w introduced for technical reasons should take values in 01] such that high frequencies are weighted down or cut o. If there exists a smooth function : ^R ^! ^R with Fourier transform ^ such that w() =^j()^ ^j², the product f²()w() can be viewed as the spectral density of the smoothed stochastic process ^R^R(t^;u)dN(u).

In Section 2, we obtain our main result on the weak convergence of ET⁽^w⁾ by proving the measurability and stochastic equicontinuity of ET⁽^w⁾ and the weak convergence of its nite dimensional distributions. As in Dahlhaus (1988), we use a proof for the stochastic equicontinuity which is based on uniform bounds for the moments of the increments of ET⁽^w⁾. However, our method of deriving these bounds is dierent as problems arise from the nonintegrability of point process spectra.

The derivation is technical and therefore put into an appendix. The conditions for measurability are stated in Theorem 2.2. This result is also valid in the case of stationary time series.

In Section 3, we give some applications of these results to the statistical analysis of point processes. In particular, we discuss parametric and nonparametric spectral density estimates obtained by maximizing an approximation to the log likelihood function.

2 Weak convergence of the empirical spectral process

For some measurable functionw :^R^!^R, let^L²w(^R) denote the space of all complex valued functions g on ^Rfor which the seminorm

w(g) =^Z

R

jg()^j²w()d¹⁼²

is nite. Further, if ^F is a subset of ^L²_w(^R), let ^X be the space of all bounded, complex valued functions on ^F which are uniformly continuous with respect to the seminorm. We equip ^X with the Borel-eld ^B^X generated by the open sets corresponding to the uniform norm^kx^k¹ = sup^jx(g)^j for x²^X.

Now, if the spectrum f² is bounded, it follows from the Cauchy-Schwarz inequality and the boundedness of I⁽^T⁾ that the sample pathsET⁽^w⁾(!) are uniformly continuous with respect to w. Therein the empirical spectral processes dier from ordinary empiricalprocesses, which in general have discontinuous sample paths. The dierence is important as we make use of continuity for proving the measurability of E_T⁽^w⁾ with respect to ^B^X.

The limit process of E_T⁽^w⁾ for T ^! ¹ is dened by its nite dimensional distributions. Therefore, we call a stochastic process E_f⁽^w²⁾ a spectral process if its sample paths are in^X almost surely and its nite dimensional distributions are normal with mean zero and

cov^fEf⁽^w²⁾(g)Ef⁽^w²⁾(h)^g

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= 2 HH²²⁴

Z

R

2g()w()h()w()f⁴(^;)dd + 2 HH²²⁴

Z

R

g()w()h()w() + h(^;)w(^;)f²()²d (2.1) wheref⁴ is the cumulant spectrum of order four of the point processN. The higher order cumulant spectra fk and the corresponding reduced cumulant measures Ck⁰

are dened analogously to (1.1) and (1.2), respectively cf. Brillinger (1972)].

For the results in this paper, we need to impose conditions on the strength of the dependence of the data and on the size of the index class ^F. The latter is determined by the covering number of ^F, which we denote by

N(w^F) = inf^fm²^Nj9g¹:::gm ²^L²w(^R)⁸g ²^F : min

1kmw(g^;gk)< ^g e.g. Pollard (1984)]. If^F is a totally bounded subset of ^L²_w(^R),N(w^F) is nite for all > 0.

Assumptions

(A1) N is an orderly, stationary point process on ^R with nite mean intensity p and reduced cumulant measures C_k⁰ such that there exists a constant C with

Z

R

k ;1(1 +^juj^j)^jdCk⁰(u¹:::uk^;1)^j C^k for allj ²^f1:::k^;1^g and k 2.

(A2) h :^R^!^Ris a Borel-measurable function of bounded variation withh(x) = 0 for allx⁶²01].

(A3) w :^R^!^R is nonnegative, bounded and^L¹-integrable.

(A4) ^F is a totally bounded subset of ^L²_w(^R) such that for all g ² ^F the product gw is bounded and the covering numbers of ^F satisfy

Z

1

0

log^fN(uw^F)²=u^g]²du <¹: We now state our main theorem.

Theorem 2.1

Suppose that Assumptions (A1) - (A4) hold. Then the empirical spectral processET⁽^w⁾(g)^,g ²^F converges weakly on^X to the spectral processEf⁽^w²⁾(g)^, g ²^F.

Proof. We will prove the stochastic equicontinuity and the measurability of the empirical spectral process and the weak convergence of its nite dimensional distributions to that of E_f⁽^w²⁾. Then the weak convergence ofE_T⁽^w⁾follows by Theorem 10.2

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in Pollard (1990), in which the outer measure^P can be replaced by the measure^P due to the measurability of ET⁽^w⁾.

For the proof of the measurability ofE_T⁽^w⁾, let (^A^P) be the underlying probability space.

Theorem 2.2

Suppose that Assumptions (A1) - (A3) hold and let ^F be a totally bounded subset of ^L²_w(^R). Then the empirical spectral processET⁽^w⁾(g)^, g ²^F ^{is a} measurable mapping into (^X^B^X).

Proof. We rst prove the measurability of the nite dimensional projections of the empirical spectral process. For xed g ²^F, we have due to dominated convergence

ET⁽^w⁾(!g) = lim_k

!1 Z k

;kg()I⁽^T⁾(!)^;f²()]w()d

pointwise for all! ². Thus, it suces to show measurabilityof the integrals on the right hand side. For this, letC^;kk] denote the space of all continuous functions on ^;kk] endowed with the topology of uniform convergence. Then the corresponding Borel-eld ^BC^;kk^] and the -eld generated by the projections t(x) = x(t) coin- cide. Hence the periodogram I⁽^T⁾ is a measurable mapping into C^;kk] since all projections I⁽^T⁾() are measurable. Further, it follows from the Cauchy-Schwarz inequality that the mapping

x ^7!^pT ^Z

;kk^]g()w()x()^;f²()]d

is continuous and thus (^BC^;kk^]^B^C)-measurable where ^B^C is the Borel-eld of ^C. This now implies the (^A^B^C)-measurability of the above integrals.

Now since^F is totally bounded,^X is separable and thus the measurability of the empirical spectral process follows from the uniform continuity of its sample paths and the measurability of its nite dimensional projections.

Note that for the uniform continuity of the sample paths it is sucient that the spectrum f² is bounded. Therefore, the assertion of the theorem holds also in the case of stationary time series under the assumptions stated in Dahlhaus (1988).

Theorem 2.3

Suppose that Assumptions (A1) - (A4) hold. Then the empirical spectral process ET⁽^w⁾(g), g ²^F is stochastically equicontinuous, i.e. for each > 0 and " > 0 there exists > 0 ^{such that}

limsup_T

!1

Pfsup^]^jET⁽^w⁾(g^;h)^j> ^g< "

where ] =^f(gh)²^F²^jw(g^;h) < ^g.

The proof of Theorem 2.3 is technical and therefore put into the Appendix. For the next theorem, we only require niteness of the integrals in Assumption (A1).

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Theorem 2.4

Suppose that Assumptions (A1) - (A3) hold. Further, let g¹:::gk ²

L

2w(^R)be such that the products gj w are bounded. Then

fE_T⁽^w⁾(gj)^gj⁼¹:::k ^!^D ^fE_f⁽^w²⁾(gj)^gj⁼¹:::k:

A similar central limit theorem has been proved in the nontapered case by Tuan (1981). However, in order to prove the same result in the tapered case, which is of much practical importance e.g. Dahlhaus (1990)], we require the concept of L⁽^T⁾- functions, which are used to deal with data tapers cf. Dahlhaus (1983, 1990)]. We therefore give a sketch of a proof, which is put into the Appendix, as we use the same techniques as in the proof of Theorem 2.3.

3 Application to the spectral analysis of point processes

In this section, we present some applications of the above results to the statistical analysis of point processes. Throughout this section, we assume that (A1) and (A2) hold.

Example 3.1

Let ^F = ^f1⁰^]^j ² 0⁰]^g and w() = 1⁰⁰^](). Then, ^F satises Assumption (A4) and we obtain a functional limit theorem for the empirical spectral distribution function on the interval 0⁰]. More generally, we can set^F =^f1D^jD ²

D g where ^D is a Vapnik-Cervonenkis class e.g. Ganssler (1983), p 22] of subsets D 0⁰] to get the same result for the empirical spectral measure ^R_Df²()d for all D ²^D.

Example 3.2

For the estimation of the covariance density of a point process, we consider

q^⁽²^w⁾(u) =^Z

R

exp(iu)I⁽^T⁾()^;p^⁽^T⁾=2 ]w()d (3.1) with symmetric and smooth weight function w such that w(0) = 1 and w() = O(^f1 +^j^jg^;3;^") for some > 0. Further, we dene q²⁽^w⁾ by (3.1) with f² and p substituted forI⁽^T⁾and ^p⁽^T⁾, respectively. Then,q⁽²^w⁾is related to the true covariance density by

q²⁽^w⁾(u) = (2 )^;1^Z

R

w(u^ ^;v)q²(v)dv (3.2) where ^w is the Fourier transform of w. Thus, the weight function in the frequency domain corresponds to a smoothing kernel in the time domain.

Let ^F = ^fgu^ju ² 0u⁰]^g with gu() = exp(iu). Since w(gu ^;gv) C^ju^;v^j for some constant C > 0, the class ^F satises Assumption (A4) and therefore by Theorem 2.1 we obtain the functional convergence

pT(^q²⁽^w⁾(u)^;q²⁽^w⁾(u)) = E_T⁽^w⁾(gu) +^pT ^w(u)2 (^p⁽^T⁾^;p) ^!^D Z⁽^w⁾(u)

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for u²0u⁰], whereZ⁽^w⁾(u) is normally distributed with mean zero and covariance cov^fZ⁽^w⁾(u)Z⁽^w⁾(v)^g = 2 HH²²⁴

Z

R

2e^iu^;^ivw()w()f⁴(^;)dd + 2 HH²²⁴

Z

R

(eⁱ⁽^u⁺^v⁾+eⁱ⁽^u^;^v⁾)w()²f²()²d + HH¹H³²

Z

R

(e^iuw(v) + e^ ^ivw(u))w()f^ ³(^;)d

+ ^w(u)^w(v)2 f²(0): (3.3)

As we can see now, the weight function w balances variance and smoothness of the estimate: As the bandwidth of the smoothing kernel in (3.2) increases, the weight function gets more concentrated and the variance decreases.

The above result can be used to derivea simultaneouscondence band for ^q²⁽^w⁾(u), u²0u⁰]. Application of the continuous mapping theorem yields

pT sup_u

20u⁰^]^jq^⁽²^w⁾(u)^;q²⁽^w⁾(u)^j^!^D _usup

20u⁰^]^jZ⁽^w⁾(u)^j:

Then ⁿ

q^⁽²^w⁾(u)T^;1⁼²z⁽^w⁾^o_u20u⁰^] (3.4) is an asymptotic simultaneous condence band, where z⁽^w⁾ denotes the upper 100 percentile point of the limit statistic supu²⁰u⁰^]^jZ⁽^w⁾(u)^j. The problem now is to obtain the distribution of sup_u²⁰_u⁰^]^jZ⁽^w⁾(u)^j, which appears to be extremely dicult. However, if we generate realisations Z¹⁽^w⁾:::Z_B⁽^w⁾ of the process Z⁽^w⁾ on a suitably ne grid, we can use the empirical distribution of supu²⁰u^0]^jZ_b⁽^w⁾(u)^j, b = 1:::B as an approximation. For this, we have to estimate the covariances (3.3). For the rst integral, a consistent estimator has been presented by Taniguchi (1982), the other integrals can be estimated similarly.

As an illustration, we apply this method to some data which describe the state of activity of a computer: An event has been recorded whenever the computer changed its state from busy to idle which was dened by the absence of any user interaction for more than ve minutes. The data set consists of 1539 events which occured in an interval of length T = 1695236s (about 20 days).

Figure 1 shows the covariance density estimate ^q²⁽^w⁾ and the corresponding simultaneous condence band (3.4) with = 0:05 for the data. Here, the weight function has been set to w() = exp(^;²²=2) with = 500s, which corresponds to a Gaussian smoothing kernel. The covariance density exhibits two signicant positive peaks: One for time lags smaller than 3h and another for a delay of about 7d. The former peak indicates that the process tends to form clusters, while the latter suggests some weekly structure in the data.

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0 1 2 3 4 5 6 7 8 9 -2.0

-1.0 0.0 1.0 2.0 3.0 4.0 5.0

Lag u [days]

qˆ2 (w) (u) [10-6 ]

Figure 1: Estimated covariance density ^q⁽²^w⁾ (solid line) with simultaneous 5% condence bands (dashed lines) for the computer data.

We now turn to the problem of spectral density estimation. Suppose that N is a stationary point process with spectral density f². Given a realization of the process on the interval 0T], we want to t a spectral density f ²^F⁰ to the data.

It is well known e.g. Brillinger (1972)] that the random variables I⁽^T⁾(2 j=T) are asymptotically independent and exponentially distributed with mean f²(2 j=T).

This suggests approximating the log likelihood function by

L (T⁾

M (f) =^;1 T

M

X

j⁼⁰

nlogf(2 j=T) + I⁽^T⁾(2 j=T) f(2 j=T)

o

and estimating f² by maximizing ^L⁽M^T⁾(f) with respect to f ² ^F⁰. This approach has been proposed by Hawkes and Adamopoulos (1973) and further discussed by Brillinger (1975) and Tuan (1981) for parametric families, for which the procedure is a point process version of a procedure suggested by Whittle (1953) for the analysis of time series.

Subsequently, we will use the following continuous version of^L⁽_M^T⁾

L

(T⁾(f) =^;^Z

R

nlogf() + I⁽^T⁾() f()

ow()d (3.5)

with w :^R ^!^R satisfying Assumption (A3) and w() = 0 for all < 0. Let f⁽^T⁾ denote a sequence of functions maximizing ^L⁽^T⁾. Substituting f² in (3.5) for I⁽^T⁾, we obtain the corresponding theoretical function ^L(f), which is maximized by f². The next theorem states the consistency of f⁽^T⁾ as an estimate for f².

Theorem 3.3

Let ^F⁰ be a subset of ^L²_w(^R) with envelope F ²^L²w(^R) andf² ²^F⁰. Then, if ^F =^ff^;1^jf ²^F⁰^g satises Assumption (A4),

w(f⁽^T⁾^;f²)^!^P 0 as T ^!¹: 8

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Proof. As in Example 3.2 in Dahlhaus (1988), we obtain from Theorem 2.1 and the Continuous Mapping Theorem cf. Pollard (1984), Theorem IV.12]

fsup^2F⁰^jL

(T⁾(f)^;^L(f)^j= sup_f

2F

0

jT^;1⁼²E_T⁽^w⁾(f^;1)^j^!^P 0

which implies^L(f⁽^T⁾)^;^L(f)^!^P 0. Now if^L(fn) converges to^L(f²) for a determin- istic sequence fn, we obtain with a Taylor expansion that w(fn^;f²) ^!0, which proves the result.

Uniform convergence off⁽^T⁾ to f⁰ requires further assumptions about ^F⁰. If, for example,^F⁰ is equicontinuous, then for any compact setK ^R

sup²K^jf⁽^T⁾()^;f²()^j^!^P 0 as T ^!¹:

In the next example, we present an explicit nonparametric function class which satises the requirements of Theorem 3.3.

Example 3.4

Consider the class^Fr(S) =^Fr(Sc⁰:::crc) of smooth functions f on S ^Rsuch that

jf⁽ⁱ⁾(x)^j ci

for 0 i r and

jf⁽^r⁾(x)^;f⁽^r⁾(y)^j c^jx^;y^j:

Further, suppose that w(x) C(1 +^jx^j)^; where > 2(r + ) + 1 and r + > 2.

Then ^Fr(^R⁺) fullls Assumption (A4). This can be seen by constructing an "- covering of ^Fr(^R⁺) from "k-coverings of ^Fr(Ik), Ik = kk + 1) with k k for some large k ² ^N. Using the uniform norm on Ik, the entropy of ^Fr(Ik) is of order O("^;1k ⁼⁽^r⁺⁾) cf. Kolmogorov and Tikhomirov (1961)]. If we choose "k ="k with r + < < (^;1)=2 and k such that 1=2"^;2⁼⁽^;1) k "^;1⁼, then "k

increases suciently fast to guarantee

logN("w^Fr(^R⁺)) J"^;^{r +}¹

Now the function class ^F = ^ff^;1^jf ² ^Fr(^R⁺) f c^g with c > 0 is itself a subset of some class ~^Fr(^R⁺) with dierent constants, and it therefore also fullls Assumption (A4).

A stronger result than Theorem 3.3 can be obtained in the case where we t a parametric model given by the class of spectral densities ^F⁰ = ^ff^j ² ^g. When dealing with parametric estimation, a point of view is to regard the parametric model only as an approximation to the true process. Therefore, we do not assume that the true spectral density f² belongs to the model class ^F⁰. If we measure the distance between a tted model specied by and the true process by ^;L(), the parameter ⁰ which maximizes^L() then determines the best approximation to the true process. By maximizing^L⁽^T⁾(), we obtain an estimate ^⁽^T⁾ for ⁰.

Assumptions

Let^F⁰ =^ff^j²^g be a parametric family.

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(B1) ^L() has an unique maximum ⁰ which is an interior point of ^R^d. (B2) There exists a compact subset of such that

liminf_T

!1

PfL(⁰)^;inf

2 C

L

(T⁾() > "⁰^g= 1

for some"⁰ > 0. Further, the functions f() are continuous in ()²^R and there exist constants c¹c² such that 0 < c¹ f() c² < ¹ for all ² and ²^R.

(B3) ^F =^ff^;1^j²^g satises Assumption (A4).

(B4) f^;1 admits continuous rst and second derivatives with respect to in a neigh- borhood U(⁰) of ⁰, denoted by the vector ^rf^;1 and the matrix ^r²f^;1, respectively. The families ^frf^;1^j ² U(⁰)^g and ^fr²f^;1^j ² U(⁰)^g satisfy Assumption (A4).

Theorem 3.5

Assume (B1) to (B3). Then we have ^⁽^T⁾^!^P ⁰. If additionally (B4)

holds, then p

T(^⁽^T⁾^;⁰)^!^D ^N(0W^;1⁰ ⁰W^;1⁰ ) where

⁰ = 2 H⁴H²^;2^Z

R 2

rf^;1⁰ ()^rf^;1⁰ ()⁰f⁴(^;)w()w()dd + 2 H⁴H²^;2^Z

R

rf^;1⁰ ()^rf^;1⁰ ()⁰f²()²w()²d and

W⁰ =^Z

R

(f²()^;f⁰())^r²f^;1⁰ ()w()d +^Z

R

rlogf⁰()^rlogf⁰()⁰w()d:

Proof. The result follows directly from Theorem 2.1. The proof is similar to that in Dahlhaus (1988).

Example 3.6

Tuan (1981) suggested to approximate the spectral density by rational functions of the form

f() = p() q() = p

2 ⁿ+a¹ⁿ^;1+::: + an

ⁿ+b¹ⁿ^;1+::: + bn

where = (pa¹:::anb¹:::bn) is the unknown parameter. Suppose the parameter space is compact. Then if p and q have no zeros in ^R⁺, the parametric class ^F⁰ satises Assumptions (B1) - (B4).

To motivate this remark, consider functions of the form g(xy) = p(x)^;yq(x) where p and q are polynomials of xed order. Since these functions form a nite dimensional vector space, the class of sets^f(xy)^jg(xy)0^gis a Vapnik-Cervonenkis (VC) class (cf. Pollard (1984), Lemma II.18). Then, the graphs G(f^;1) form also a VC-class and the Approximation Lemma (cf. Pollard (1984), Lemma II.25) yields (B3). The conditions on the derivatives are checked similarly.

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Appendix

A key role in our proof of Theorem 2.3 is played by the function L⁽^T⁾ : ^R ^! ^R, T ²^R⁺ which is given by

L⁽^T⁾() :=

( T ^j^j 1=T

1=^j^j ^j^j> 1=T : (A.1) A similar, but periodic function was introduced in Dahlhaus (1983) as a tool for handling the cumulants of discrete time series statistics. The above function L⁽^T⁾ is the corresponding version for time-continuous stochastic processes. Its properties are summarized in the following lemma.

Lemma A.1

^Suppose w fullls Assumption (A3). Further, let ² ^R ^and p²^N. We obtain with constants Kp independent of T:

(i) L⁽^T⁾() is monotone increasing in T ²^R⁺ and decreasing in ²^R⁺. (ii) L⁽^T⁾(c) c^;1L⁽^T⁾() for all c²(01].

(iii)L⁽^T⁾( + )L⁽^T⁾(^;) L⁽^T⁾(⁺²)L⁽^T⁾( ^;) + L⁽^T⁾( + )L⁽^T⁾(⁺²). (iv) ^Z

R

L⁽^T⁾()w( + )d K¹log(T) for T e. (v) ^Z

R

L⁽^T⁾( + )L⁽^T⁾(^;)w( + )d K¹log(T)L⁽^T⁾( + ) for T e. (vi) ^Z

R

L⁽^T⁾()^pd KpT^p^;1^. (vii)^Z

R

L⁽^T⁾( + )^pL⁽^T⁾(^;)^pd KpT^p^;1L⁽^T⁾( + )^p.

Proof. The proofs are straightforward and similar to those in Dahlhaus (1983).

But unlike its periodic counterpart, the function L⁽^T⁾ is not ^L¹-integrable, and the inequalities (iv) and (v) therefore require an ^L¹-integrable weight functionw.

Using the denition ofL⁽^T⁾, we can now derive an upper bound for the Fourier transform of a data taper. Let V (h) denote the total variation of the function h.

Then if h is of bounded variation, simple calculations yield the inequality

Z

R

jh(t + u¹)h(t + uk)^;h(t)^k^jdt ^kh^k^k¹^;1V (h)(^ju¹^j+::: +^juk^j): (A.2) From this, we obtain for the Fourier transform of h⁽^T⁾

jHk⁽^T⁾()^j 1=2^Z

R

jh(t)^k^;h(t^; =)^k^jdt 1=2^kh^k^k¹^;1V (h)k ^j^j^;1: On the other hand, we have ^jHk⁽^T⁾()^j ^kh^k^k¹T. Hence, we obtain as an upper bound

jHk⁽^T⁾()^j K^kL⁽^T⁾() (A.3) for allk ²^N and a constant K independent of T.

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For sake of simplicity, we assume for the proof of Theorem 2.3 that the mean intensityp is known and therefore replace d⁽^T⁾() by

~d⁽^T⁾() =^Z

R

h⁽^T⁾(t)exp(^;it)dN(t)^;pdt]:

At the end of the proof, we indicate the modications needed for the case that p is estimated by ^p⁽^T⁾.

The next lemma, which is a slightly stronger version of Theorem 4.1 in Brillinger (1972), gives an approximation for the cumulants of ~d⁽^T⁾().

Lemma A.2

Under Assumptions (A1) and (A2), there exists a constant c > 0such that for all ¹:::k ²^R and k 2

jcum^f~d⁽^T⁾(¹)::: ~d⁽^T⁾(k)^g^;(2 )^k^;1H_k⁽^T⁾(¹ ++k)fk(¹:::k^;1)^j c^k:

Proof. The lemma is an immediate consequence of relation (A.2) and Assumption (A1).

Note that because of Assumption (A1) we can choose the constant c such that also

jfk(¹:::k^;1)^j c^k for allk 2.

Proof of Theorem 2.3. We start by proving that there exists a constantc⁰ such that

jcumk^fET⁽^w⁾(g)^gj (2k)!c^k⁰w(g)^k (A.4) for all K ² ^N where cumk^fE_T⁽^w⁾(g)^g denotes the k-th cumulant of E_T⁽^w⁾(g). Using Lemma A.2 we obtain

jcum¹^fET⁽^w⁾(g)^gj ^pT ^Z

R

jg()^jw()^jEI⁽^T⁾()^;f²()^jd

c²^f2 H²^pT^g^;1w(g)w(1): (A.5) For k2, we nd

jcumk^fE_T⁽^w⁾(g)^gj

2 H²⁽^T⁾(0)]^;^kT^k=²^Z

R k

jg(¹)^jw(¹)^jg(k)^jw(k)

jcum^f~d⁽^T⁾(¹)~d⁽^T⁾(^;¹)::: ~d⁽^T⁾(k)~d⁽^T⁾(^;k)^gjd¹dk: (A.6) In order to apply the product theorem for cumulants cf. Brillinger (1981), Theorem 2.3.2], let ^Pi:p: denote the sum over all indecomposable partitionsP¹:::Pm of the

table ¹ ^;¹

... ...

k ^;k

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with pj = ^jPj^j 2 as ^Ef~d⁽^T⁾()^g = 0. Further if Pj = ^fj¹:::jp^j^g, we write

"j =j¹+::: + jp^j. Then using Lemma A.2, we nd for the cumulant in (A.6)

jcum^f~d⁽^T⁾(¹)~d⁽^T⁾(^;¹)::: ~d⁽^T⁾(k)~d⁽^T⁾(^;k)^gj

X

i:p:

m

Y

j⁼¹

n(2 )^p^j;1^jHp⁽^T^j⁾("j)^jjfp^j(j¹:::jp^j^;1)^j+c^p^j^o which, by using fk(¹:::k^;1) c^k and (A.3), is less than

X

i:p:(2 )²^kc²^kK²^k ^X

JM

Y

j²JL⁽^T⁾("j) (A.7) where M =^f1:::m^g. Substituting (A.7) in (A.6), we obtain as an upper bound

X

i:p:

X

JM(2 c²K²H²^;1)^kT^;^k=²^Z

R k

k

Y

j⁼¹^jg(j)^jw(j)^Y

j²JL⁽^T⁾("j)d¹dk: For J =, the integral is equal to

Z

R k

jg()^jw()d^k w(g)^kw(1)^k:

Similarly, if m = 1 and J = ^f1^g, the integral is bounded by Tw(g)^kw(1)^k since L⁽^T⁾("¹) =T. For J ⁶= and m > 1, we split ^f1:::k^g into disjoint sets I and I^C and J into disjoint sets J⁰ and J⁰^C to be selected later. Using the Cauchy-Schwarz inequality, the integral now is less than

Z

R k

Y

j²I^jg(j)^j²w(j) ^Y

j²I^Cw(j) ^Y

j²J⁰^CL⁽^T⁾("j)²d¹dk

1=²

Z

R k

Y

j²I^C

jg(j)^j²w(j)^Y

j²Iw(j) ^Y

j²J⁰L⁽^T⁾("j)²d¹dk

1=²

: (A.8) Now we have to make a suitable choice for I, I^C,J⁰ and J⁰^C. Since J is not empty, we can dene J⁰ = ^fj⁰^g for some arbitrary j⁰ ² J. Then there exists i⁰ such that i⁰ or ^;i⁰ is in Pj⁰, and we can set I =^fi⁰^g. We obtain with Lemma A.1 (vi) and (vii) for the rst integral in (A.8)

Z

R

jg(i⁰)^j²w(i⁰)^Z

R k ;1

Y

i²I^Cw(i) ^Y

j²J⁰^CL⁽^T⁾("j)² ^Y

i²I^Cdi

di⁰

Z

R

jg(i⁰)^j²w(i⁰)ⁿ^kw^k^j¹^J^j;1w(1)²⁽^k^;j^J^j)(KT)^j^J^j;1^odi⁰

= w(g)²ⁿ^kw^k^j¹^J^j;1w(1)²⁽^k^;j^J^j)(KT)^j^J^j;1^o

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