A functional large deviations principle for quadratic forms of Gaussian stationary processes

Statistics & Probability Letters 43 (1999) 299–308

A functional large deviations principle for quadratic formsof Gaussian stationary processes

F. Gamboaa;b, A. Rouaultc, M. Zanid ; ∗

aFacult�e de Math�ematiques et d’Informatique, Universit�e de Picardie Jules Verne, 33 rue St. Leu, F-80039 Amiens Cedex 01,France

bEcole Polytechnique, C.M.A.P., F-91128 Palaiseau Cedex, FrancecD�epartement de Math�ematiques, Batiment Fermat, Universit�e de Versailles F-78035, Versailles, France

dLab. de Statistiques, Universit�e Paris-Sud, F-91405 Orsay Cedex, France

Received June 1998

Abstract

A functional large deviations principle is proved for quadratic forms of centered stationary Gaussian processes indexedby discrete or continuous time. c© 1999 Elsevier Science B.V. All rights reserved

MSC: 60F10; 11E25; 60G15; 47B35

Keywords: Large deviations; Quadratic forms; Gaussian processes; Toeplitz matrices; Wiener–Hopf operators

1. Introduction

Let X = (Xn)n∈Z be a discrete time stationary Gaussian process and Y = (Ys)s∈R be a continuous timestationary Gaussian process. Let gX and gY denote the spectral densities of X and Y de�ned respectively onT= [−�; �] and R. They are non-negative integrable even functions and the covariance of both processesmay be computed as the Fourier transforms of these functions:

E(XnXn+k) =12�

∫ �

−�eiktgX (t) dt; (n; k ∈ Z); (1)

E(YsYs+r) =12�

∫ ∞

−∞eirtgY (t) dt; (s; r ∈ R): (2)

In what follows, we will consider the random positive measure built with quadratic forms of X :

� Xn (f) =1nX (n)∗f(�n)X (n); f∈C([mgX ;MgX ]); (3)

∗ Corresponding author. E-mail: [email protected].

0167-7152/99/$ – see front matter c© 1999 Elsevier Science B.V. All rights reservedPII: S0167 -7152(98)00270 -3

300 F. Gamboa et al. / Statistics & Probability Letters 43 (1999) 299–308

where X (n)∗= (X1; : : : ; Xn); �n is the covariance matrix of X (n) and C([mgX ;MgX ]) denotes the set of allcontinuous functions on [mgX ;MgX ] = [ess-inf gX , ess-sup gX ]. In Section 2.1 we recall the meaning of thematrix f(�n) (see the proof of Lemma 2). In the continuous case, let �T be the covariance operator ofY (T ) = (Ys; s ∈ [0; T ]). It is a trace-class operator from HT = L2([0; T ]) to itself. Its spectrum �(�T ) lies in[0; MgY ] where MgY = ess-sup gY . For any continuous function on [0; MgY ] we may de�ne the operator f(�T )belonging to L(HT ) (see Aupetit, 1991). Let

�YT (f) :=1T

∫ T

0Ys[f(�T )Y (T )] (s) ds=

1T〈Y (T ); f(�T )Y (T )〉; (4)

where 〈 ; 〉 is the scalar product in L2([0; T ]).The aim of this note is to establish a large deviations principle (LDP) for the random measures � Xn and

�YT . For the sake of completeness we recall the de�nition of a LDP (cf. Dembo and Zeitouni, 1993).

De�nition 1. We say that a sequence (Rn) of probability measures on a measurable Hausdor� space (U;B(U ))satis�es a LDP with rate function I if1. I is lower semi-continuous, with values in R+ ∪ {+∞}.2. For any measurable set A of U :

−I(int A)6 lim infn→∞

1nlogRn(A)6 lim sup

n→∞1nlogRn(A)6− I(cloA);

where I(A) = inf �∈A I(�).We say that the rate function I is good if its level sets {x ∈U : I(x)6a} are compact for any a¿0.

More generally, a sequence of U -valued random variables is said to satisfy a LDP if their distributionssatisfy a LDP.

For a �xed f, a LDP for � Xn (f) is already available since it is a particular case of Proposition 3 of Bercuet al. (1997). One interest of this paper is to present a functional version of this LDP. We give here largedeviations principles for random measures with convex but not strictly convex rate functions. It is substantiallydi�erent from the classical Sanov’s case. The main idea here is to adapt the proof of the Baldi’s theorem(Dembo and Zeitouni, 1993).The paper is organized as follows: in Section 2.1 (resp. Section 2.2) we establish a LDP for (� Xn ) (resp.

for (�YT )). All technical proofs are postponed to Section 2.3.

2. Main results

2.1. Discrete time case

From now on, we write g; �n; m and M for gX ; � Xn ; mgX and MgX respectively.We assume that the function g is continuous and positive on T, so that m¿ 0. Let M([m;M ]) be the set

of all positive bounded measures on [m;M ]. Endowed with the weak topology, it is Polish (metric separableand complete). We �rst check that (3) de�nes a random positive measure on [m;M ] satisfying a law of largenumbers. This is the aim of the following lemma.

Lemma 2. Let �n1; : : : ; �nn denote the eigenvalues of �n.

F. Gamboa et al. / Statistics & Probability Letters 43 (1999) 299–308 301

1. Almost surely �n is in M([m;M ]) and there exist independent �2(1)-distributed random variablesZn1 ; : : : ; Z

nn such that

�n =1n

n∑i=1

�ni Zni ��ni : (5)

2. For any f in C([m; M ]),

�n(f)→ �(f) in probability when n→ +∞;where

�(f) =∫[m; M ]

f(t) t dP(t);

and P denotes the image probability of the normalized Lebesgue measure on the torus by the applicationg, so that

�(f) =12�

∫ �

−�g(x)f[g(x)] dx:

Proof. Let O be an orthogonal n × n matrix such that O∗�nO is the diagonal matrix whose ith diagonalelement is �ni . Let f be in C([m;M ]), recall that f(�n) is de�ned by f(�n) = ODfO∗ where Df is thediagonal matrix whose ith diagonal element is f(�ni ).From the Cochran theorem, we may write �n(f) as

�n(f) =1n

n∑i=1

�ni Zni f(�

ni ); (6)

where for n¿1, the random variables Zn1 ; : : : ; Znn are independent �

2(1)-distributed and do not depend on f.Consequently,

�n =1n

n∑i=1

�ni Zni ��ni ;

so that 1 of the Lemma 2 follows.Set Pn the empirical measure

Pn =1n

n∑i=1

��ni : (7)

From the theorem of Szeg�o on Toeplitz forms (see Grenander and Szeg�o, 1958; Avram, 1998),

Pn =⇒n→+∞P; (8)

where ⇒ denotes the weak convergence. Moreover, for all n¿1, the support of Pn is contained in [m;M ].

nVar(�n(f)) =2n

n∑i=1

�2i; nf(�i; n)2−−−−−→n→+∞

1�

∫Tg(t)2f[g(t)]2 dt;

which proves 2.

302 F. Gamboa et al. / Statistics & Probability Letters 43 (1999) 299–308

Now we deal with the large deviations properties of (�n). For any f in C([m;M ]) set

�(f) =−∫[m; M ]

log(1− 2tf(t))2t

d�(t) if ∀ t ∈ [m;M ]; tf(t)612;

�(f) =+∞ otherwise: (9)

and for any � in M([m;M ]) de�ne the convex dual function of � by

�∗(�) = supf∈C([m; M ])

(∫f(t)�(dt)− �(f)

): (10)

The following lemma gives another expression for �∗. It is a consequence of Theorem 5 of Rockafellar(1971).Let for �¿ 0,

(�) = 12 (�− 1− log �):

The function is convex (it is the Cramer transform of the �2(1) distribution). Its recession function (seeRockafellar, 1971) is �→ �=2.

Lemma 3. �∗ is a good convex rate function. For any � in M([m;M ]) having, with respect to �, theLebesgue decomposition � = l�+ �, then

�∗(�) =∫[m; M ]

(l(t))t

d�(t) +∫[m; M ]

d�(t)2t

whenever the integrals are de�ned ;

�∗(�) =+∞ otherwise: (11)

When the integrals are de�ned, �∗ can also be rewritten as

�∗(�) =12�

∫]−�; �]

(l ◦ g(t)) dt +∫[m; M ]

d�(t)2t

:

The main result of this section follows.

Theorem 4. (�n) satis�es a LDP with good rate function �∗.

Corollary 5. Almost surely �n ⇒ �.

Remark. Formula (11) shows that the case m = 0 is more delicate. Actually, it may be tackled as in thecontinuous time case.

Proof. A result analogous to Theorem 4 was established in another framework in Gamboa and Gassiat (1997)with both time-invariant discretization and random weights (see also Dembo and Zeitouni, 1996; Cattiaux andGamboa, 1998).The proof is based on the ideas of Baldi’s theorem (Theorem 4.5.20 of Dembo and Zeitouni, 1993). The

framework will not exactly be the same, as the limit normalized cumulant will not be de�ned everywhere buta careful study of the exposed points shows that the proof can be adapted. For any function f in C([m;M ]),the normalized cumulant generating function of �n is

�n(f) =1nlogE(exp[n�n(f)]) =

1nlogE

n∏j=1

exp(�njf(�nj )Z

nj )

:

F. Gamboa et al. / Statistics & Probability Letters 43 (1999) 299–308 303

There are three cases to consider for the asymptotic behavior of �n(f).Case 1: ∀ t ∈ [m;M ]; t f(t)¡ 1=2;

�n(f) =− 12n

n∑j=1

log(1− 2�njf(�nj ));

and therefore from (8),

limn→+∞�n(f) =−

∫[m; M ]

log(1− 2tf(t))2t

d�(t) = �(f):

Case 2: ∃t ∈ [m;M ] such that t f(t)¿ 1=2. For n large enough, there exists i in {1; : : : ; n} such that�ni f(�

ni )¿ 1=2 and �n(f) = +∞. Thereforelim

n→+∞�n(f) = �(f) = +∞:Case 3: ∀t ∈ [m;M ]; tf(t)61=2 and ∃t ∈ [m;M ] such that tf(t) = 1=2. We do not know in general what

happens for the asymptotic behavior of �n(f). Nevertheless it does not matter: to �nd a LDP for �n, wewill show that it is enough to consider functions in cases 1 or 2, since they are dense in the set of exposinghyperplanes.Upper bound: The upper bound holds for any compact set of M([m;M ]) in view of Theorem 4.5.3 b) of

Dembo and Zeitouni, 1993 and the following lemma, whose proof is postponed to Section 2.3.

Lemma 6. For all �¿ 0 and � in M([m;M ]), there exists f� in C([m;M ]) such that tf�(t)¡ 1=2 for all tand ∫

f�(t)�(dt)− �(f�)¿min{�∗(�)− �; 1

�

}: (12)

Exponential tightness: For all a¿ 0,{sup

‖f‖∞61�n(f)¿a

}⊂{�n(1)¿a}:

Fix � in ]0; 1=4M [, the classical Chernov bound and (8) give

lim supn

1nlogP(�n(1)¿a)6− �a− 1

2

∫[m; M ]

log(1− 2�x) dP(x);

and

lima→+∞ lim supn

1nlogP(�n(1)¿a) =−∞:

Hence the sequence (�n) is exponentially tight, and the upper bound holds for any closed set of M([m;M ]).Lower bound: We study the set of the exposed points of �∗. We recall that � in M([m;M ]) is an exposed

point of �∗ if there exists a function f of C([m;M ]) such that for all � in M([m;M ]) with � 6= �,∫[m; M ]

f(t)�(dt)− �∗(�)¿∫[m; M ]

f(t)�(dt)− �∗(�): (13)

The function f is called the exposing hyperplane.Denote by F the set of the absolutely continuous measures with respect to � having a positive continuous

Radon–Nikodym derivative. Then F is a subset of M([m;M ]), and the following Lemma shows that F isa set of exposed points of �∗:

304 F. Gamboa et al. / Statistics & Probability Letters 43 (1999) 299–308

Lemma 7. Let � = l� be in F. Set

f(t) =12t

(1− 1

l(t)

); t ∈ [m;M ]:

Then � is an exposed point of �∗ with exposing hyperplane f. Furthermore, for some ¿ 1, we have�( f)¡+∞.

The proof is given in Section 2.3. Now, from the following lemma, F is a dense subset of M([m;M ])and this implies the lower bound.

Lemma 8. Let � be in M([m;M ]) such that �∗(�)¡+∞. There exists a sequence of positive functions (ln)in C([m;M ]) such that ln�⇒ � and limn→+∞�∗(ln�) = �∗(�).

The proof is given in Section 2.3.

2.2. Continuous-time case

From now on, we write g; M and �T for gY ; MgY and �YT , respectively.

We assume that g is continuous on R, and we follow the scheme of the discrete time case. Let M([0; M ])be the set of the positive bounded measures on [0; M ] endowed with the weak topology, and let C([0; M ])be the set of the continuous functions on [0; M ]. The eigenvalues {�(T )k }k¿1 of �T satisfy 0¡�(T )k 6M and∑

k¿1 �(T )k ¡+∞.

Lemma 9.1. Almost surely �T de�ned by (4) is a positive bounded measure and there exists a sequence{Z (T )k }k¿1 ofindependent �2(1) distributed random variables such that

�T =1T

∞∑k=1

Z (T )k �(T )k ��(T )k:

2. For any f in C([0; M ]),

�T (f)→ �(f) in probability as T → +∞;where

�(f) =∫ +∞

−∞g(y)f(g(y))

dy2� :

Proof. Let (e(T )k ) be a complete orthonormal system of eigenvectors of �T with associated eigenvalues {�(T )k },then

Y (T ) =∑k¿1

〈Y (T ); e(T )k 〉e(T )k ;

f(�T )Y (T ) =∑k¿1

f(�(T )k )〈Y (T ); e(T )k 〉e(T )k ;

�YT (f) =1T

∑k¿1

�(T )k f(�(T )k )ZTk ;

F. Gamboa et al. / Statistics & Probability Letters 43 (1999) 299–308 305

where ZTk = (�(T )k )−1〈Y (T ); e(T )k 〉2 are independent and �2(1) distributed. So we have,

�YT =1T

∞∑k=1

ZTk �(T )k ��(T )k

:

This de�nes a positive �nite measure since∑∞

k=1 �(T )k is �nite. From Ginovian (1994)

QT =1T

∞∑k=1

�(T )k ��(T )k=⇒T→+∞

� (14)

(see also Bryc and Dembo, 1997 and the seminal book of Grenander and Szeg�o, 1958, p. 139). To show theconvergence of �T (f), we notice that

TE[�T (f)− E�T (f)]2 → 1�

∫g(x)2f(g(x))2 dx:

By analogy with the discrete-time case, for any function f in C([0; M ]) set

�(f) =−∫[0; M ]

log(1− 2tf(t))2t

d�(t) if ∀ t ∈ [0; M ]; tf(t)612

�(f) =+∞ otherwise: (15)

For any � in M([0; M ]), de�ne the Fenchel–Legendre dual of �:

�∗(�) = supf∈C([0; M ])

(∫f(t)�(dt)− �(f)

): (16)

In view of Rockafellar (1971), we have the following lemma:

Lemma 10. �∗ is a good convex rate function. Let � in M([0; M ]) having, with respect to �, the Lebesguedecomposition � = l�+ �. If t → (l(t))=t is in L1(�) and 1=t is in L1(�) then

�∗(�) =∫[0; M ]

(l(t))t

d�(t) +∫[0; M ]

d�(t)2t

=12�

∫R (l ◦ g(y)) dy +

∫[0; M ]

d�(t)2t

:

Otherwise, �∗(�) = +∞.

Theorem 11. (�T ) satis�es a LDP with good rate function �∗.

The proof of this theorem is similar to the one of Theorem 4. Remark that for all f in C([0; M ]) such thattf(t)¡ 1=2, the normalized cumulant generating function of �T on f is

�T (f) =−∫[0; M ]

log(1− 2tf(t))2t

dQT (t)

which, from (14) converges towards �(f) as T goes to +∞.The analogues of Lemmas 7 and 8 are a little di�erent here, since there is a problem of integrability in 0.

306 F. Gamboa et al. / Statistics & Probability Letters 43 (1999) 299–308

Denote by G the subset of M([0; M ]) of measures � = l� such that l is in C([0; M ]) and satis�es thecondition

(A) limt→0

l(t) = 1; limt→0(1− l(t))=t is �nite:

∀t ∈ [0; M ] ; l(t)¿ 0:

Lemma 12.1. Let � = l� be in G. Set

f(t) =12t

(1− 1

l(t)

); t ∈ [0; M ]:

Then � is an exposed point of �∗ with exposing hyperplane f.2. Let � be inM([0; M ]) such that �∗(�)¡+∞. There exists (ln) in C([0; M ]) such that ln�⇒ �; limn→+∞�∗(ln�) = �∗(�) and for all n; ln satis�es (A).

To complete the proof of Theorem 11, it remains to show the exponential tightness. It is a classical Cherno�bound; the di�erence with the discrete time case is that we integrate on the tapered measure QT .

2.3. Proofs of Lemmas 6; 7; 8 and 12

Proof of Lemma 6. From the de�nition of �∗, for all �¿ 0, there exists f in C([m;M ]) such that (12)holds. In case tf(t)61=2, we may add some �¡ 0 so that (12) holds with another �.

Proof of Lemma 7. The proof is based on the strict convexity of the function : for two non-negative realnumbers x; y such that x 6= y

(x)− (y)¡ (x − y) ′(x) (17)

and

′(x) =12

(1− 1

x

):

Integrating the relation (17), we obtain (13).In addition, for any � in F the associated exposing hyperplane f is strictly bounded by 1=(2t). Conse-

quently, for some ¿ 1, we have �( f)¡+∞.

Proof of Lemma 8. This lemma has been proved in the unpublished paper (Gamboa and Gassiat, 1991). Forsake of completeness, we recall this proof.Easy considerations on the properties of this function lead to

(�+ �′)6 (�) +�′

2(�¿ 0; �′¿0): (18)

Let l and l be non-negative measurable functions on [m;M ]. Integrating (18),

�∗((l+ l)�)6�∗(l�) +12

∫l(t) d� (19)

whenever the terms on the right are de�ned.

F. Gamboa et al. / Statistics & Probability Letters 43 (1999) 299–308 307

Step 1: If �= l�+ �, with �∗(�)¡∞; l non-negative and continuous, and the singular part � such that 1=tin L1(�). Since P has full support on [m;M ], there exists a sequence (hn) of positive functions in C([m;M ])such that hn d�⇒ �=t. From the lower semi-continuity of �∗,

lim infn→+∞ �∗((l+ thn)�)¿�∗(�):

From inequality (18),

�∗((l+ thn)�)6�∗(l�) +12

∫[m; M ]

hn d�;

and then

lim infn→+∞ �∗((l+ thn)�)6�∗(�):

We now show that the lemma is true if � = l � with l �-a.s. non-negative integrable.Step 2: We prove the result for � = l � assuming that l is integrable and that for some �¿ 0; l¿ � �-a.s.

There exists a sequence (ln) of continuous positive functions such that ln converges both in L1(�) norm and�-a.s. to l and ln ¿ �=2. Since on ]�=2;+∞[ the function is Lipschitzian we claim that the lemma holds.Step 3: De�ne l�:=lIl¿�+ �Il6�. Apply second step and inequality (19) noticing that l� converges in L1(�)

to l and that l�¿l.Step 4: For � = l�+ �, combine �rst and third step.

Proof of Lemma 12.1. For l in C([0; M ]) satisfying (A), �∗(l �)¡∞ because t→ (l(t))=t is continuous, and f(t)=(1−1=l(t))=2tis in C([0; M ]).

2. We prove part 2 in two steps:Step 1: This step is the same as the one of Lemma 8. Remark that l+ thn satis�es condition (A).Step 2: Take � = l� with l and (l)=t in L1(�).Fix �¿ 0. We approximate l by l4 satisfying (A) with the following construction:

• Let l1 be the function de�ned by l1(x) = 1 if x∈ [0; �]; l(x) if x∈ [�;M ]; where � is chosen so that|�∗(l1�)− �∗(l�)|6� and ‖ l1 − l ‖1 6� (the norm ‖ : ‖1 is related to the measure �).

• Set l2 = l1Il1¿�+ �Il16�, where � is chosen such that |�∗(l2�)−�∗(l1�)|6� and ‖ l2− l1 ‖1 6�. Thereforel2¿�.

• Set l3 = 1 on [0; �] and l3 is positive continuous on [�;M ], with l3¿�=2 and ‖ l3 − l2 ‖1 6�.• Let l4 be in C([0; M ]), de�ned by

l4(x) = 1 if x ∈ [0; �];l4(x) = is linear on [�; �];

l4(x) = l3(x) if x ∈ [�;M ]with � chosen such that |�∗(l4�)− �∗(l3�)|6� and ‖ l4 − l3 ‖1 6�.

Therefore |�∗(l4�)− �∗(l�)|64� and ‖ l4 − l ‖1 64�.

Acknowledgements

The authors want to thank B. Bercu for helpful discussions.

308 F. Gamboa et al. / Statistics & Probability Letters 43 (1999) 299–308

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