Table Of ContentSome convergence results on quadratic forms for
random fields and application to empirical
0
1 covariances
0
2
n Frédéric LAVANCIER Anne PHILIPPE
a
J
Université de Nantes, Laboratoire de Mathématiques Jean Leray
8
]
T Abstract
S
Limit theorems are proved for quadratic forms of Gaussian random
.
h fields in presence of long memory. We obtain a non central limit theo-
t rem under a minimal integrability condition, which allows isotropic and
a
m anisotropic models. We apply our limit theorems and those of Ginovian
(1999) to obtain the asymptotic behavior of the empirical covariances of
[
Gaussian fields, which is a particular example of quadratic forms. We
3 showthatitispossibletoobtainaGaussian limitwhenthespectralden-
v sity is not in L2. Therefore the dichotomy observed in dimension d = 1
6 betweencentralandnoncentrallimittheoremscannotbestatedsoeasily
6 dueto possible anisotropic strong dependencein d>1.
1 Keywords: Gaussian Random field ; Long memory ; Non central
4
limit theorems ; Quadratic mean convergence ; Stochastic integral ; Tri-
.
2 angular array.
1
8
0 1 Introduction
:
v
i Many statistical methods proposed for long memory processes are based on
X
statistics, which can be expressed using quadratic forms (see Beran, 1994, for
r a review). The empirical covariance sequence (see Hosking, 1996; Hannan,
a
1976),theestimationofthelongmemoryparameterusingWhittlecontrast(see
Fox and Taqqu,1986;Giraitis and Surgailis,1990;Giraitis and Taqqu,1999,for
example)orusingtheintegratedperiodogram(seeLobato I. and Robinson P.M.,
1996),somechangepointdetectionprocedure(seeBeran and Terrin,1996)rely
on quadratic forms.
It would be interesting to validate the same statistical tools for the study of
random fields having long memory. As a first step, we study in this paper the
convergence of the quadratic forms in the d−dimensional (d>1) case.
Let us introduce our framework. Let X = (Xn)n∈Zd be a stationary L2
randomfield havinglong memory,i.e. its covariancesequence (r(n))n∈Zd is not
a summable series. We assume that X is a Gaussian process and admits the
1
linear representation
X = a(x)ei<n,x>dW(x), n∈Zd, E =[−π,π]d, (1)
n
ZE
where W is the Gaussian white noise spectral measure and where the function
a is in L2(E). < ·,· > denotes the inner product in Rd. Hereafter, we denote
by f the spectral density of X, f is proportional to |a(x)|2, and we have
r(n)=fˆ := ei<n,λ>f(λ)dλ. (2)
n
ZE
The quadratic forms associated to X and (gi)i∈Nd ∈ℓ1(Nd) are defined as
1
Q = g X X (3)
n nd i−j i j
iX∈AnjX∈An
where A = {1,...,n}d. The statistics Q can be rewritten as a functional of
n n
the periodogramI of X. Indeed let
n
I (t)=(2πn)−d X X ei<k−l,t>,
n k l
k,Xl∈A2n
we have
Q =(2π)d g(t)I (t)dt, (4)
n n
ZE
where
g(t)=(2π)−d g e−i<j,t>.
j
jX∈Zd
In dimension one (d = 1), many papers deal with the asymptotic behav-
ior of Q for long memory processes. When the intensity of the memory is
n
not too strong or when specific conditions are imposed to g in order to kill
the effect of the long memory involved by f, the convergence rate and the
asymptoticnormalityobtainedinshortmemoryarepreserved. The asymptotic
normality of the quadratic forms is proved by Avram (1988), Fox and Taqqu
(1987)andGinovian and Sahakian(2005)intheGaussiancaseandextendedby
Giraitis and Surgailis (1990) to include the non-Gaussian linear case. Avram’s
result is restricted to f and g in L2, but requires only integrability conditions.
On the opposite, Fox and Taqqu (1987), Ginovian and Sahakian (2005) specify
the behavior of f and g around zero. The extension to linear models obtained
by Giraitis and Surgailis (1990) holds under the assumptions of Avram (1988)
or Fox and Taqqu (1987).
Under other conditions on f and g fixing the behavior of both functions
aroundzero,Rosenblatt(1961),Fox and Taqqu(1985),Terrin and Taqqu(1990)
prove non central limit theorems for Gaussian processes. The normalization is
notstandard(i.e. itisnotequalton−d/2)andthe limitingprocessisnotGaus-
sian. The extension to linear models is proved in Giraitis et al (1998) in the
general context of bivariate Appell polynomials.
2
Fixing the behavior of f and g around zero, the dichotomy in d=1 can be
summarize as follows. Assume that
f(x)∼|x|2α when x→0,
g(x)∼|x|2β when x→0.
The limiting distribution depends on the value of (α+β).
1
• If α+β >− then the limit is Gaussian and the normalization is n−1/2
4
(Q satisfies a central limit theorem).
n
1
• If α+β <− then the limiting distributionis notGaussianandthe nor-
4
malization in this case is n2(α+β), which is lower than n−1/2 (Q satisfies
n
a non central limit theorem).
1
The case α+β = is investigated in Ginovian and Sahakian (2005) (up to a
4
slowly varying function in f and g which vanishes at zero). In this case, the
limit remains gaussian.
In the d-dimensional case (d > 1), a central limit theorem has been proved
by Ginovian (1999), in the same spirit as Avram (1988) (i.e. f and g are in
L2 and satisfy an integrabilitycondition). This result is recalledin Section 2.1.
Whenf org arenotinL2,somecentrallimittheoremshavebeenprovedunder
some specific conditions on these functions, see Theorem 3.1 in Doukhan et al.
(1996) and Lemma 3.3 in Boissy et al. (2005).
On the other hand, few results are available on non-central limit theorems
in dimension d > 1. In Doukhan et al. (1996), the asymptotic of Q is inves-
n
tigated when the memory of X is "isotropic" (see Lavancier (2005) Definition
1), i.e. the memory of X is due to |x|2α in the expression of f where |·| de-
notesthe Euclidiannorm. The authorsobtainanon-centrallimit theoremvery
similar to Fox and Taqqu (1985). However the investigation of "anisotropic"
memory and more general functions g (including the particular case of the em-
pirical covariance) is of great interest for statistical applications. In Lavancier
(2005),differentclassesofrandomfieldshavinglongmemoryaredescribed: The
important role of anisotropic processes in terms of modeling is illustrated.
Our paper is organized as follows. In Section 2, after a reminder of the
central limit theorem obtained by Ginovian (1999), we present a non-central
limittheoremunderageneralintegrabilityconditiononf andg. We showthat
this condition is satisfied by a large class of isotropic and anisotropic models
(including all previous results). Since random fields in d>1 may exhibit much
more different types of long memory than processes in d = 1, the dichotomy
betweencentralandnoncentrallimittheoremsisnotsosimpletoformulate. As
already highlighted in Remark 4.2 of Dobrushin and Major (1979), a spectral
singularityoutsidezeromayleadtonewlimitingresults. Thisfactisillustrated
through examples in Section 2.3.
3
InSection3,weapplylimittheoremsforquadraticformstotheconvergence
oftheempiricalcovarianceseries. Contrarytotheclassicalsettingindimension
one, we show that it is possible to obtain a Gaussian limit in dimension d > 1
when f is not in L2.
2 Convergence of quadratic forms
2.1 Central limit theorem
We recallin this sectionthe centrallimit theoremobtained by Avram(1988) in
dimension d = 1 and extended by Ginovian (1999) to d > 1. This result will
be useful in section 3, where the asymptotic behavior of the covarianceseries is
investigated.
Theorem 1 (Avram (1988), Ginovian (1999)). Let X be a random field as in
(1) and denote f its spectral density. Let Q , defined in (3), be the quadratic
n
form associated with g and X. If f ∈ Lp and g ∈ Lq with p ≥ 2, q ≥ 2 and
1/p+1/q≤1/2, then
nd/2(Q −E(Q ))−L→N 0,2(2π)3d f2(t)g2(t)dt , (5)
n n
(cid:18) ZE (cid:19)
L
where −→ denotes the convergence in law.
Thisresultprovidesalargesettingsincenospecificformforfunctionsf and
g isrequired,butonlyintegrabilityconditions. Inreturn,f andg arerestricted
toL2. AsimilarresultisobtainedinDoukhan et al.(1996),Th. 6.2.,butunder
semi-parametric assumptions on f and g.
When f or g are not in L2 but a compensation between them occurs, a
central limit theorem may still arise as in Fox and Taqqu (1985) for d = 1.
To our knowledge, two situations in dimension d > 1 have been investigated
so far. In dimension d = 2, assuming f and g have a tensorial product form
around zero, Boissy et al. (2005) obtain a central limit theorem, which can be
straightforwardly extended to d > 2. In Doukhan et al. (1996), Th. 6.1, the
authors prove the same result when f has an isotropic form around zero and
when g is regular enough and g(0)=0.
Itisknownthatthelonelycondition0< f2(t)g2(t)dt<∞isnotsufficient
E
to obtain a central limit theorem, as proved in Ginovian and Sahakian (2005),
R
Prop. 2.2. A generalsetting leading to a centrallimit theorem(included f org
not in L2) is still missing. This issue is not investigated in this paper.
2.2 Main result : non-central limit theorem
The following theorem gives the convergence of the quadratic forms under the
general condition (H) below on f and g. Contrary to the previous studies in
theonedimensioncaseorindimensiond>1,thisconditiondoesnotrequireto
4
specify the behavior of f around its singularities. Such a condition is satisfied
by many examples as shown in Section 2.3.
Let ζ be a function defined on Rd, hereafter we denote by ζ (p ∈ N)
(p)
the periodic function with period 2πp (with respect to each component), that
coincides with ζ on the set pEd,
ζ (x)=ζ(x) ∀x∈[−pπ,pπ]d. (6)
(p)
Theorem 2. Let X be the linear field defined by (1) and Q the quadratic form
n
defined by (3).
Assume that for all x ∈ E a(−x) = a(x), and g(−x) = g(x). Moreover,
assume that
a(x)=a˜ (x)L (x),
(1) 1
(g(x)=g˜(1)(x)L2(x),
where a˜ is a homogeneous function of degree α (i.e. ∀c > 0, ∀x ∈ E, a˜(cx) =
cαa˜(x)), where g˜ is a homogeneous function of degree 2β and where L and L
1 2
are bounded functions, continuous at zero and L (0)6=0 (i=1,2).
i
If the following assumption (H) is satisfied
2
d
1
(H): a˜2(x)a˜2(y) |g˜(t)| dt dxdy <∞,
ZZR2d "ZRd k=1(1+|xk+tk|)(1+|yk−tk|) #
Y
then
nd+2α+2β(Q −E(Q ))−L→2
n n
L2(0)L (0) a˜(x)a˜(y) g˜(t)H(x+t)H(y−t)dtdW(x)dW(y), (7)
1 2
ZZR2d ZRd
where H(z)= d eizj−1.
j=1 izj
ItseemsthaQt(H)isalmostnecessaryfortheconvergence,sinceitguarantees
that the integral in (7) exists.
The proof is relegated to Section 2.4. The main tool consists in rewriting
Q as a double stochastic integral. Then the convergence in quadratic mean is
n
deducedfromasimpleconvergenceinL2(R2d). Notethatcontrarytotheproofs
for non central limit theorems in Terrin and Taqqu (1990) and Doukhan et al.
(1996), we do not apply the scheme of convergence of Dobrushin and Major
(1979),especiallytheirLemma3whichinvolvesaspectralmeasureconvergence
assumption.
2.3 Some class of models satisfying the hypothesis (H)
The setting of random fields allows many kind of dependencies (see Lavancier
(2005)). Apart from its intensity, the dependence can be isotropic or can occur
all over several particular directions, depending on the form of the spectral
5
density. Since we did not want to restrict ourselves to one particular case,
Theorem 2 involves a general integrability condition, i.e. assumption (H). We
checkinthissectionthatthishypothesisisnottoorestrictive. Indeed,weprove
that in all the preceding studies about quadratic forms of Gaussian fields, this
condition is fulfilled. Moreover, it allows to extend to new cases, mainly when
the dependence is not isotropic.
As a consequence of the following Lemmas and according to Theorem 2, a
non-central theorem for Q arises for all the cases treated in this section.
n
Lemma 3. Assume that there exists some positive constants c and c′ such that
|a˜(x)|≤c d |x |α/d
i=1 i (8)
( |g˜(x)|≤c′Qdi=1|xi|2β/d.
If moreover α>−d/2, β >−d/2 and αQ+β <−d/4, then (H) is fulfilled.
Remark 1. The result of our Theorem 2 in the setting of Lemma 3 is the same
as the convergence stated in dimension 1 by Terrin and Taqqu (1990) (Th. 1).
The assumptions in Lemma 3 allow the filter a(x) to be isotropic, i.e. equiv-
alent at zero, up to a constant, to |x|α. This is the hypothesis done in Theorem
5.1 of Doukhan et al. (1996) where, moreover, it is assumed β =0.
Proof. We have to check that the function defined on (Rd)4 by
a˜2(x)a˜2(y)|g˜(t)||g˜(s)|
(x,y,s,t)7→
d (1+|x +t |)(1+|y −t |)(1+|x +s |)(1+|y −s |)
k=1 k k k k k k k k
is integrable onQ(Rd)4. From the hypothesis of Lemma 3, it is enough to prove
that for all k =1,...,d
|x |2α/d|y |2α/d|t |2β/d|s |2β/d
k k k k
(t ,s ,x ,y )7→
k k k k
(1+|x +t |)(1+|y −t |)(1+|x +s |)(1+|y −s |)
k k k k k k k k
isintegrableonR4. SuchintegralsarestudiedinLemma1ofTerrin and Taqqu
(1990): UndertheassumptionsofLemma3onαandβ,thisintegralisfinite.
Whenthefunctions aandg involvedinTheorem2donotsatisfy(8), condi-
tion(H) may be investigatedthanks to power countingtheorems (cf. Theorem
2 in Terrin and Taqqu (1990)). The following lemma focus on a particular sit-
uation in dimension d = 2: When a˜ admits one or two independent lines of
singularities.
Lemma 4. Assume that d = 2 and that g˜ follows the same conditions as in
(8).
If, for p6=q,
a˜(x ,x )=|x +px |αp|x +qx |αq,
1 2 1 2 1 2
and if α >−1/2, α > −1/2, β >−1 and α +α +β <−1/2, then (H)
p q p q
is fulfilled.
6
Proof. Checking (H) is equivalent to prove that
|x1+px2|2αp|x1+qx2|2αq|y1+py2|2αp|y1+qy2|2αq|t1|β|t2|β|s1|β|s2|β
dtdsdxdy
ZR8 2k=1(1+|xk+tk|)(1+|yk−tk|)(1+|xk+sk|)(1+|yk−sk|)
(9)
Q
is finite. We apply Theorem 2 of Terrin and Taqqu (1990). Let us introduce
some notations. The integral above can be written :
|L (u)|2αp|L (u)|2αq|L (u)|2αp|L (u)|2αq×
1 2 3 4
R8
Z
16
|L (u)|β|L (u)|β|L (u)|β|L (u)|β (1+|L (u)|)−1du, (10)
5 6 7 8 k
k=9
Y
where u = (x ,x ,y ,y ,t ,t ,s ,s ) and the L ’s are the linear functionals
1 2 1 2 1 2 1 2 k
involved in (9). For instance L (u)=x +px , L (u)=t , L (u)=x +t .
1 1 2 5 1 9 1 1
Let T ={L ,...,L } and let γ be the exponent associated to L in (10).
1 16 k k
For instance, γ = 2α , γ = β, γ = −1. Consider now the subsets W ⊂ T
1 p 5 9
such that span(W)∩T = W. A subset W is said padded if any L ∈ W is a
k
linear combination of the L ’s in W −{L }.
i k
The integrability of (9) near 0 is obvious since 2α > −1, 2α > −1 and
p q
β >−1. According to Theorem 2 in Terrin and Taqqu (1990), the integrability
atinfinityisachievedifforeverypaddedW consideredabovebutT,d (W):=
∞
rank(T)−rank(W)+ γ <0.
T−W k
Themaximumvalueford (W)isobtainedwithW ={L ,...,L }. Inthis
∞ 9 16
P
case d (W)=8−6+4α +4α +4β. This leads to α +α +β <−1/2.
∞ p q p q
2.4 Proof of Theorem 2
Proof. LetZ beaGaussianrandomspectralmeasureassociatedtothemeasure
µ. Major (1981) defines
f(x,y)dZ(x)dZ(y)
ZZR2d
for all f ∈ H where H denotes the space of functions f: Rd×Rd → C such
µ µ
that f(−x,−y) = f(x,y) and |f(x,y)|2dµ(x)dµ(y) < ∞. The second order
moment of this integral satisfies, for all f in H ,
µ
R
2
E f(x,y)dZ(x)dZ(y) ≤2 |f(x,y)|2dµ(x)dµ(y). (11)
(cid:18)ZZ (cid:19) ZR2d
And the so-called Ito formula follows: For all f and f in H ,
1 2 µ
f (x)f (y)dZ(x)dZ(y)= f (x)dZ(x) f (y)dZ(y)− f (x)f (x)dµ(x).
1 2 1 2 1 2
ZZ Z Z Z
(12)
7
Now, let us rewrite Q as a double stochastic integral. According to (1),
n
1
Q −E(Q )= g (X X −r(l−k))
n n nd k−l k l
kX∈AnlX∈An
1
= g(t) ei<k−l,t>(X X −r(l−k)) dt. (13)
nd k l
ZE " kX∈AnlX∈An #
From the Ito formula (12)
X X −r(l−k)
k l
= a(x)ei<k,x>dW(x) a(y)ei<l,y>dW(y)− ei<l−k,x>a2(x)dµ(x)
ZE ZE ZE
= a(x)a(y)ei(<k,x>+<l,y>)dW(x)dW(y),
ZZE2
where µ, the spectral measure of ǫ, is proportional to the Lebesgue measure λ.
Assume that g(−x)=g(x) and a(−x)=a(x), then the function
1
(x,y)7→a(x)a(y) g(t) ei<k,x+t> ei<l,y−t>dt (14)
nd
" ZE kX∈An lX∈An #
belongs to H . Therefore we can rewrite (13) as
λ
1
Q −E(Q )= a(x)a(y) g(t)H (x+t)H (y−t)dt dW(x)dW(y),
n n nd n n
ZZE2 (cid:20) ZE (cid:21)
(15)
where H (t)= ei<k,t>.
n k∈An
In (15), we make the change of variables x → x/n, y → y/n, t → t/n.
Since the GausPsian measure W satisfies, for all Borelian set A, W(n−1A) =
n−1/2W(A) and since a˜ and g˜ are homogeneous, we get
nd+2α+2β(Q −E(Q ))=
n n
x y
a˜ (x)a˜ (y)L L ψ (x,y)dW(x)dW(y),
(n) (n) 1 n 1 n n
ZZnE2 (cid:16) (cid:17) (cid:16) (cid:17)
where
t 1 x+t 1 y−t
Ψ (x,y)= g˜ (t)L H H dt.
n (n) 2 n nd n n nd n n
ZnE (cid:18) (cid:19) (cid:18) (cid:19) (cid:18) (cid:19)
As a consequence, according to (11), it suffices, for proving (7), to show that
the following integral tends to zero as n→∞:
2
x y
a˜ (x)a˜ (y)L L Ψ (x,y)−a˜ (x)a˜ (y)L2(0)Ψ(x,y) dxdy
(n) (n) 1 n 1 n n (n) (n) 1
ZZnE2(cid:20) (cid:16) (cid:17) (cid:16) (cid:17) (cid:21)
8
where
Ψ(x,y)=L (0) g˜(t)H(x+t)H(y−t)dt.
2
ZRd
From the decomposition A B −CD = (A −C)B +(B −D)C, the L2-norm
n n
above is lower than the sum 2(I +I ) where
1 2
2
I = a˜ (x)a˜ (y)Ψ (x,y)−a˜ (x)a˜ (y)Ψ(x,y) ×
1 (n) (n) n (n) (n)
ZZnE2(cid:20) (cid:21)
2
x y
L L dxdy
1 1
n n
(cid:20) (cid:16) (cid:17) (cid:16) (cid:17)(cid:21)
and
2 2
x y
I = L L − L2(0) a˜ (x)a˜ (y)Ψ(x,y) dxdy.
2 1 n 1 n 1 (n) (n)
ZZnE2(cid:20) (cid:16) (cid:17) (cid:16) (cid:17) (cid:21) (cid:20) (cid:21)
The following lemma will be useful.
Lemma 5.
d
1 z 1
(i) ∀ z =(z ,...,z )∈nE, H ≤ π 1∧ .
1 d nd n n |z |
(cid:12)(cid:12) (cid:16) (cid:17)(cid:12)(cid:12) jY=1 (cid:18) j (cid:19)
(cid:12) (cid:12)
(cid:12) (cid:12) d
1
(ii) ∀ z =(z ,...,z )∈Rd, |H(z)|≤ 2 1∧ .
1 d
|z |
j=1 (cid:18) j (cid:19)
Y
1 z
(iii) For a.e. z ∈Rd, lim 1I (z) H −H(z) =0.
n→∞ nE nd n n
(cid:12) (cid:12)
(cid:12) (cid:16) (cid:17) (cid:12)
Proof of Lemma 5. Since for all j, |z |≤nπ,(cid:12) (cid:12)
j (cid:12) (cid:12)
(cid:12)(cid:12)n1dHn(cid:16)nz(cid:17)(cid:12)(cid:12)=jY=d1(cid:12)(cid:12)n1eiznj eeiizznjj −−11(cid:12)(cid:12)=jY=d1(cid:12)(cid:12)sinz(jz/j2/2)(cid:12)(cid:12)(cid:12)(cid:12)sinz(jz/j2/n2n)(cid:12)(cid:12)
(cid:12) (cid:12) d (cid:12) (cid:12) (cid:12) (cid:12)(cid:12) (cid:12)
(cid:12) (cid:12) (cid:12) 1 (cid:12) (cid:12) (cid:12)(cid:12) (cid:12)
≤ π 1∧ .
|z |
j=1 (cid:18) j (cid:19)
Y
Similarly, for all z ∈Rd,
d eizj −1 d sin(zj/2) d 1
|H(z)|= = ≤ 2 1∧ .
iz z /2 |z |
j=1(cid:12) j (cid:12) j=1(cid:12) j (cid:12) j=1 (cid:18) j (cid:19)
Y(cid:12) (cid:12) Y(cid:12) (cid:12) Y
(cid:12) (cid:12) (cid:12) (cid:12)
Finally, for proving(cid:12)(iii), su(cid:12)ppose fi(cid:12)rst that d(cid:12)=1. If z 6=0,
(cid:12)n1dHn nz −H(z)(cid:12)=(cid:12)n1einz eeiinzz −−11 − eiziz−1(cid:12)≤ |z2|(cid:12)n(eeiinznzi−z1) −1(cid:12).
(cid:12) (cid:16) (cid:17) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12)
(cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12)
(cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12)
9
The norm in the right hand side term is equivalent to |z|/n when n goes to
infinity, hence it tends to 0. In dimension d, (iii)is provedby induction thanks
to the decomposition AB−CD =(A−C)B+(B−D)C.
Let us first prove that I asymptotically vanishes. Since L is bounded, it
1 1
suffices to prove the convergence to 0 of
2
I = a˜2 (x)a˜2 (y) Φ(11)(t,x,y)dt dxdy, (16)
11 (n) (n) n
ZZnE2 (cid:20)ZnE (cid:21)
with
t 1 x+t 1 y−t
Φ(11)(t,x,y)=g˜ (t) L −L (0) H H
n (n) 2 n 2 nd n n nd n n
(cid:18) (cid:18) (cid:19) (cid:19) (cid:18) (cid:19) (cid:18) (cid:19)
and
2
I = a˜2 (x)a˜2 (y)L (0)2 Φ(12)(t,x,y)dt dxdy, (17)
12 (n) (n) 2 n
ZZnE2 (cid:20)ZnE (cid:21)
with
1 x+t 1 y−t
Φ(12)(t,x,y)=g˜ (t) H H −H(x+t)H(y−t)
n (n) nd n n nd n n
(cid:20) (cid:18) (cid:19) (cid:18) (cid:19) (cid:21)
In both I and I , the 2π-periodicity of g, H and H allows us to reduce
11 12 n
the domain of integration nE (with respect to t) to
nD ={|x−t|<nπ}∩{|y+t|<nπ}∩nE.
x,y
Therefore, (i) of Lemma 5 can be applied and since L is bounded,
2
d
1 1
Φ(11)(t,x,y)dt ≤c |g˜ (t)| 1∧ 1∧ dt,
n (n) |x +t | |y −t |
(cid:12)ZnE (cid:12) ZnDx,y k=1(cid:18) k k (cid:19)(cid:18) k k (cid:19)
(cid:12) (cid:12) Y
(cid:12) (cid:12)
w(cid:12) here c is a positive(cid:12)constant. Hence,
2
d
1
I ≤c′ a˜2(x)a˜2(y) |g˜(t)| dt dxdy.
11
ZZR2d "ZRd k=1(1+|xk+tk|)(1+|yk−tk|) #
Y
Besides, according to (i) of Lemma 5,
1I (x,y)1I (t )1I (t )a˜2 (x)a˜2 (y)Φ(11)(t ,x,y)Φ(11)(t ,x,y)
nE2 nDx,y 1 nDx,y 2 (n) (n) n 1 n 2
(cid:12) (cid:12)
(cid:12) (cid:12)
(cid:12) t t (cid:12)
(cid:12)(cid:12) ≤c a˜2(x)a˜2(y)|g˜(t1)||g˜(t2)| L2 1 −L2(0) L2 2 −L2(0)(cid:12)(cid:12) .
n n
(cid:12) (cid:18) (cid:19) (cid:12)(cid:12) (cid:18) (cid:19) (cid:12)
(cid:12) (cid:12)(cid:12) (cid:12)
FromthecontinuityofL at0,thisterm(cid:12) tendstozerofor(cid:12)a(cid:12)nyfixed(x,y,t ,t(cid:12))∈
2 (cid:12) (cid:12)(cid:12) 1 2(cid:12)
R4d. Therefore, thanks to assumption (H), the Lebesgue’s dominated conver-
gence theorem applies and lim I =0.
n→∞ 11
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