Rosenblatt distribution subordinated to gaussian random fields with long-range dependence

0.39 MB·English

by  N. N. Leonenko

ROSENBLATT DISTRIBUTION SUBORDINATED TO GAUSSIAN
RANDOM FIELDS WITH LONG-RANGE DEPENDENCE
N.N. LEONENKO,M.D. RUIZ-MEDINA,ANDM.S. TAQQU
6
1 Abstract. The Karhunen-Lo`eveexpansion and the Fredholm determinant formula are used,
0 to derive an asymptotic Rosenblatt-type distribution of a sequence of integrals of quadratic
2 functions of Gaussian stationary random fields on Rd displaying long-range dependence. This

n distributionreducestotheusualRosenblattdistributionwhend=1.Severalpropertiesofthis
u newdistributionareobtained. Specifically,itsseriesrepresentation,intermsofindependentchi-
J squaredrandomvariables,isestablished. ItsL´evy-Khintchinerepresentation,andmembership

8 to the Thorin subclass of self-decomposable distributions are obtained as well. The existence
and boundednessof its probability density then follow as adirect consequence.

]
T
S
Keywords. Fredholmdeterminant;Hermitepolynomials;infinitedivisibledistributions;multiple
.
h
Wiener-Itˆo stochastic integrals; non-central limit theorems; Rosenblatt-type distribution.
t
a
m AMS subject classifications. 60F99; 60E10; 60G15; 60G60.
[

4 1. Introduction
v
7 The aim of this paper is to derive and study the properties of the limit distribution, as
4 T , of the random integral
2 −→ ∞
2
0 1
. (1.1) S = (Y2(x) 1)dx,
1 T d −
0 T ZD(T)
5 where the normalizing function d is given by
T
1
v: (1.2) dT = Td−α (T), 0< α < d/2,
L
i
X
with being a positive slowly varying function at infinity, that is
L
r
a (1.3) lim (T x )/ (T) = 1,
T L k k L
→∞
for every x > 0, and D(T) Rd denotes a homothetic transformation of a set D Rd, with
k k ⊂ ⊂
center at the point 0 D, and coefficient or scale factor T > 0. In the subsequent development,
∈
D is assumed to be a regular bounded domain, whose interior has positive Lebesgue measure,
and with boundaryhaving nullLebesguemeasure. Here, Y(x), x Rd is a zero-mean Gauss-
ianhomogeneousandisotropicrandomfieldwithvaluesin{R,displa∈ying}long-rangedependence.
That is, Y is assumed to satisfy the following condition:
Condition A1. The random field Y(x), x Rd is a measurable zero-mean Gaussian homo-
{ ∈ }
geneous and isotropic mean-square continuous random field on a probability space (Ω, ,P),
A
Date: 27 May 2016.
This work has been supported in part by project MTM2015–71839–P (co-funded by European Regional De-
velopment Funds), MINECO, Spain. This research was also supported under Australian Research Council’s
DiscoveryProjects fundingscheme(projectnumberDP160101366), andunderCardiffIncomingVisitingFellow-
shipSchemeandInternationalCollaboration SeedcornFund. MuradS.TaqquwassupportedinpartbytheNSF
grants DMS-1007616 and DMS-1309009 at Boston University.
1
2 N.N.LEONENKO,M.D.RUIZ-MEDINA,ANDM.S.TAQQU
with EY2(x) = 1, for all x Rd, and correlation function E[Y(x)Y(y)] = B( x y ) of the
∈ k − k
form:
( z )
(1.4) B( z ) = L k k , z Rd, 0 < α < d/2.
k k z α ∈
k k
From Condition A1, thecorrelation B of Y is acontinuous function of r = z . Itthenfollows
k k
that (r)= (rα), r 0. Note that the covariance function
L O −→
1
(1.5) B( z ) = , 0 < β 2, γ > 0,
k k (1+ z β)γ ≤
k k
is a particular case of the family of covariance functions (1.4) studied here with α = βγ, and
(1.6) ( z ) = z βγ/(1+ z β)γ.
L k k k k k k
The limit random variable of (1.1) will be denoted as S . The distribution of S will be
∞ ∞
referred to as the Rosenblatt-type distribution, or sometimes simply as the Rosenblatt distribu-
tion because this is how it is known in the case d= 1. In that case, a discretized version in time
of the integral (1.1) first appears in [35], and the limit functional version is considered in [42]
in the form of the Rosenblatt process. In this classical setting, the limit of (1.1) is represented
by a double Wiener-Itˆo stochastic integral (see [12]; [43]). Other relevant references include, for
example, [2],[3], [15], [17], [24], [36], to mention just a few. The general approach considered
here for deriving the weak-convergence to the Rosenblatt distribution is inspired by [42], which
is based on the convergence of characteristic functions. This approach has also been used, re-
cently, in [24], to study the characteristic functions of quadratic forms of strongly-correlated
Gaussian random variables sequences.
We suppose here d 2, and thus consider integrals of quadratic functions of long-range
≥
dependence zero-mean Gaussian stationary random fields. We pursue, however, a different
methodology than in the case d = 1, which was based on the discretization of the parameter
space. Adirectextension ofthesetechniques isnotavailable whend 2. Insteadofdiscretizing
≥
the parameter space of the random field, we focus on the characteristic function for quadratic
formsfor Hilbert-valued Gaussian randomvariables (see, for example, [11]), andtake advantage
of functional analytical tools, like the Karhunen-Lo`eve expansion and the Fredholm determi-
nant formula, to obtain the convergence in distribution to a limit random variable S with
∞
Rosenblatt-type distribution.
The double Wiener-Itˆo stochastic integral representation of S in the spectral domain leads
∞
to its series expansion in terms of independent chi-squared random variables, weighted by the
eigenvaluesoftheintegraloperatorintroducedinequation(3.1)below. Theasymptoticsofthese
eigenvalues is given in Corollary 2. The infinitely divisible property of S is then obtained as
∞
a direct consequence of the previous results derived, in relation to the series expansion of S ,
∞
and the asymptotic properties of the eigenvalues. We also prove that the distribution of S is
∞
self-decomposable, and that it belongs, in particular, to the Thorin subclass. Theexistence and
boundedness of the probability density of S then follows.
∞
The outline of the paper is now described. In Section 2, we recall the Karhunen-Lo`eve ex-
pansion, introduce the Fredholm determinant formula, and use the referred tools to obtain the
characteristic function of (1.1). In Section 3, we prove the weak convergence of (1.1) to the ran-
domvariableS withaRosenblatt-typedistribution. ThedoubleWiener-Itˆostochasticintegral
∞
representation of S , its series expansion in terms of independent chi-square random variables,
∞
and the asymptotics of the involved eigenvalues are established in Section 4. These results are
applied in Section 5 to derive some properties of the Rosenblatt distribution, e.g., infinitely
divisible property, self-decomposability, and, in particular, the membership to the Thorin sub-
class. Appendices A-C provide some auxiliary results and the proofs of some propositions and
corollaries.
NON-CENTRAL LIMIT THEOREMS 3
In this paper we consider the case of real-valued random fields. In what follows we use the
symbols C,C ,M ,M , etc., to denote constants. The same symbol may be used for different
0 1 2
constants appearing in the text.
2. Karhunen-Lo´eve expansion and related results
Thissection introduces somepreliminary definitions, assumptionsandlemmas hereafter used
in the derivation of the main results of this paper. We start with the Karhunen-Lo`eve Theorem
for a zero-mean second-order random field Y(x), x K Rd , with continuous covariance
function B (x,y) = E[Y(x)Y(y)], (x,y) {K K ∈Rd ⊂Rd, d}efined on a compact set K of
0
Rd (see Section 3.2 in [1]). This theorem∈prov×ides⊂the fo×llowing orthogonal expansion of the
random field Y :
∞
Y(x) = λ φ (x)η , x K,
j j j ∈
j=1
Xp
λ φ (x) = B (x,y)φ (y)dy, k N , φ ,φ = δ , i,j N ,
k k ZK 0 k ∈ ∗ i j L2(K) i,j ∈ ∗
(2.1) (cid:10) (cid:11)
where η = 1 Y(x)φ (x)dx, for each k 1, and the convergence holds in the L2(Ω, ,P)
k √λk K k ≥ A
sense. The eigenvalues of B are considered to be arranged in decreasing order of magnitude,
R 0
that is, λ λ λ λ .... The orthonormality of the eigenfunctions φ , j N ,
leads to t1he≥un2co≥rr·e·la·t≥ionko−f1t≥he rkan≥dom variables η , j N , with variance one, sincej ∈ ∗
j ∈ ∗
E[η η ] = B (x,y)φ (y)φ (x)dydx = λ φ (x)φ (x)dx = λ δ ,
j k 0 j k j j k j j,k
ZKZK ZK
with δ denoting the Kronecker delta function. In the Gaussian case, they are independent.
For each T > 0, let us fix some notation related to the Karhunen-Lo`eve expansion of the re-
striction tothesetD(T)ofGaussian randomfieldY,withcovariance function(1.4). ByR
Y,D(T)
we denote the covariance operator of Y with covariance kernel B (x,y) = E[Y(x)Y(y)],
0,T
x,y D(T), which, as an operator from L2(D(T)) onto L2(D(T)), satisfies
∈
R (φ )(x) = B (x,y)φ (y)dy = λ (R )φ (x), l N ,
Y,D(T) l,T 0,T l,T l,T Y,D(T) l,T
ZD(T) ∈ ∗
where, in the following, by λ (A) we will denote the kth eigenvalue of the operator A. In partic-
k
ular, λ (R ) and φ respectively denote the eigenvalues and eigenfunctions
{ k,T Y,D(T) }∞k=1 { k,T}∞k=1
of R , for each T > 0. Note that, as commented, B refers to the covariance function of
Y,D(T) 0,T
Y(x), x D(T) as a function of (x,y) D(T) D(T), which, underCondition A1, defines
{ ∈ } ∈ ×
a non-negative, symmetric and continuous kernel on D(T), satisfying the conditions assumed
in Mercer’s Theorem. Hence, the Karhunen-Lo`eve expansion of random field Y holds on D(T),
and its covariance kernel B also admits the series representation
0,T
∞
(2.2) B (x,y) = λ (R )φ (x)φ (y), x,y D(T),
0,T j,T Y,D(T) j,T j,T ∈
j=1
X
where the convergence is absolute and uniform (see, for example, [1], pp.70-74). The orthonor-
mality of the eigenfunctions φ yields
{ l,T}∞l=1
1 1 ∞
(2.3) Y2(x)dx = λ (R )η2 .
d d j,T Y,D(T) j,T
T ZD(T) T j=1
X
4 N.N.LEONENKO,M.D.RUIZ-MEDINA,ANDM.S.TAQQU
In the derivation of the limit characteristic function of (1.1), we will use the Fredholm de-
terminant formula of a trace operator. Recall first that a positive operator A on a separable
Hilbert space H is a trace operator if
(2.4) A Tr(A) (A A)1/2ϕ ,ϕ < ,
k k1 ≡ ≡ ∗ k k H ∞
Xk D E
whereA denotes the adjoint of A and ϕ is an orthonormal basis of the Hilbert space H (see
∗ { k}
[34], pp. 207-209). A sufficient condition for a compact and self-adjoint operator A to belong
to the trace class is ∞k=1λk(A) < ∞. For each finite T > 0, the operator RY,D(T) is in the
trace class, since from equation (2.2), applying the orthonormality of the eigenfunction system
φ , j N , and kPeeping in mind that B (0) = 1, we have
{ j,T ∈ ∗} 0,T
∞
(2.5) Tr(R ) = λ (R ) = B (x,x)dx = dx= Td D < ,
Y,D(T) j,T Y,D(T) 0,T
| | ∞
j=1 ZD(T) ZD(T)
X
where D denotes the Lebesgue measure of the compact set D. Note that the class of compact
| |
and self-adjoint operators contains the class of trace and self-adjoint operators. Hence, under
Condition A1, from equation (2.5), the restriction of Y to D(T) admits a Karhunen-Lo´eve
expansion, convergent in the mean-square sense (i.e., in the L2(Ω, ,P)-sense), for any T > 0,
A
and for an arbitrary regular bounded domain D. Furthermore, for any k 1,
≥
(2.6) Rk f(x)= B∗(k)(x,y)f(y)dy, f L2(D(T)),
Y,D(T) 0,T ∈
ZD(T)
(k)
where B∗ denotes
0,T
B0∗,(T1)(x,y) = B0,T(x,y), k = 1,
(k) (k 1)
(2.7) B0∗,T (x,y) = B0∗,T− (x,z)B0,T(z,y)dz, k = 2,3,....
ZD(T)
From equations (2.2) and (2.7), applying the orthonormality of φ , j N , one can obtain
j,T ∈ ∗
(2.8) Tr(RYk,D(T))= j∞=1λkj,T(RY,D(T)) = ZD(T)B0∗,(Tk)(x,x)dx < ∞, k ∈N∗,
X
since, for every k 1, λ (R ) M λ (R )k = M λ (Rk ), for some positive
≥ | k Y,D(T) | ≤ | k Y,D(T) | | k Y,D(T) |
constant M. In particular, in the homogeneous random field case,
Tr(RYk,D(T)) = ∞ λkj,T(RY,D(T))= B0∗,(Tk)(xk,xk)dxk
j=1 ZD(T)
X
k 1
−
= ... B (x x ) B (x x )dx ...dx ,
0,T j+1 j 0,T 1 k 1 k
 −  −
ZD(T) ZD(T) j=1
Y
(2.9)  
and, in the homogeneous and isotropic case, for k = 2,
∞ 2( x y )
(2.10) Tr(R2 )= λ2 (R ) = L k − k dydx.
Y,D(T) j,T Y,D(T) x y 2α
j=1 ZD(T)ZD(T) k − k
X
ThefollowingdefinitionintroducestheFredholmdeterminantofanoperatorA,asacomplex-
valued function which generalizes the determinant of a matrix.
NON-CENTRAL LIMIT THEOREMS 5
Definition 1. (see, for example, [39], Chapter 5, pp.47-48, equation (5.12)) Let A be a trace
operator on a separable Hilbert space H. The Fredholm determinant of A is
∞ TrAk ∞ ∞ ωk
(2.11) (ω) = det(I ωA) = exp ωk = exp [λ (A)]k ,
l
D − − k − k
! !
k=1 k=1 l=1
X XX
for ω C, and ω A < 1. Note that Am A m, for A being a trace operator.
∈ | |k k1 k k1 ≤ k k1
Lemma 1. Let Y(x), x D Rd be an integrable and continuous, in the mean-square
sense, zero-mean,{Gaussian∈rando⊂m fie}ld, on a bounded regular domain D Rd containing the
⊆
point zero. Then, the following identity holds:
∞
E exp iξ Y2(x)dx = (1 2λ (R )iξ) 1/2 = ( (2iξ)) 1/2
j Y,D − −
− D
(cid:20) (cid:18) ZD (cid:19)(cid:21) j=1
Y
1 ∞ (2iξ)m
(2.12) = exp Tr(Rm ) ,
2 m Y,D
!
m=1
X
for R 2iξ < 1, as given in Definition 1.
Y,D 1
k k | |
Proof. Thecovariance operatorR ofY,acting onthespaceL2(D),isinthetraceclass. From
Y,D
Definition 1, the following identities hold:
∞
E exp iξ Y2(x)dx = E exp iξ λ (R )η2
  j Y,D j
(cid:20) (cid:18) ZD (cid:19)(cid:21) j=1
X
  
∞ ∞
= E exp iξλ (R )η2 = (1 2λ (R )iξ) 1/2 = ( (2iξ)) 1/2
j Y,D j − j Y,D − D −
j=1 j=1
Y (cid:2) (cid:0) (cid:1)(cid:3) Y
1/2
∞ (2iξ)m − 1 ∞ (2iξ)m
= exp Tr(Rm ) = exp Tr(Rm ) ,
− m Y,D 2 m Y,D
" !# !
m=1 m=1
X X
(2.13)
where the last two identities in equation (2.13) are finite for ξ < 1 , from the Fredholm
| | 2D
| |
determinant formula (2.11). Note that
∞ ∞
(2.14) Tr(Rm )= λm(R ) λm 1(R ) λ (R )= λm 1(R ) R < .
Y,D j Y,D ≤ 1 − Y,D j Y,D 1 − Y,D k Y,Dk1 ∞
j=1 j=1
X X
(cid:3)
Remark 1. Similarly to equation (2.12), one can obtain the following identities, which will be
used in the subsequent development: For a homothetic transformation D(T) of D Rd, with
⊂
center at the point 0 D, and coefficient T > 0,
∈
∞
E exp iξ Y2(x)dx = (1 2λ (R )iξ) 1/2 = ( (2iξ)) 1/2
j,T Y,D(T) − T −
− D
" ZD(T) !# j=1
Y
1 ∞ (2iξ)m
(2.15) = exp Tr(Rm ) ,
2 m Y,D(T)
!
m=1
X
whereλ (R ) λ (R ) λ (R ) ...,with, asbefore, λ (R ), j
1,T Y,D(T) 2,T Y,D(T) j,T Y,D(T) j,T Y,D(T)
≥ ≥ ··· ≥ ≥ { ∈
N denoting the system of eigenvalues of the covariance operator R of Y, as an operator
Y,D(T)
∗}
from L2(D(T)) onto L2(D(T)). The last identity in equation (2.15) holds for R 2iξ <
Y,D(T) 1
k k | |
1, i.e., for Tr(R )2iξ = Td D 2iξ < 1, or equivalently for ξ < 1 .
Y,D(T) | | | || | | | 2Td D
| |
6 N.N.LEONENKO,M.D.RUIZ-MEDINA,ANDM.S.TAQQU
3. Weak convergence of the random integral S
T
This section provides the weak convergence of the random integral (1.1) to a Rosenblatt-type
distribution, in Theorem 2. This results is based on the asymptotic behavior of the eigenvalues
of the integral operator (see Theorem 1 below)
α
K
1
(3.1) (f)(x) = f(y)dy, f Supp( ), 0 < α < d,
Kα x y α ∀ ∈ Kα
ZD k − k
with Supp(A) denoting the supportof operator A. Operator (3.1) can be related with the Riesz
potential ( ∆) β/2 of order β, 0 < β < d, on Rd, formally defined as (see [41], p.117)
−
−
1
(3.2) ( ∆) β/2(f)(x) = x y d+βf(y)dy,
− −
− γ(β) Rdk − k
Z
where ( ∆) denotes the negative Laplacian operator, and
−
πd/22βΓ(β/2) 1
(3.3) γ(β)= = , 0 < β < d.
Γ d−β c(d,β)
2
(cid:16) (cid:17)
Indeed, except a constant, the function (1/ x y α) in equation (3.1) defines the kernel of the
k − k
Riesz potential ( ∆)(α d)/2 of order β = (d α), for 0 < α < d. Similarly, 1/ x y 2α is
−
− − k − k
the kernel of the Riesz potential ( ∆)α d/2 of order β = (d 2α) on Rd, for 0< α < d/2.
Recall that the Schwartz space−(Rd)−is the space of of in−finitely differentia(cid:0)ble functions(cid:1)on
S
Rd, whose derivatives remain bounded when multiplied by polynomials, i.e., whose derivatives
are rapidly decreasing. Particularly, C (D) (Rd), with C (D) denoting the infinitely
0∞ ⊂ S 0∞
differentiable functions with compact support contained in D.
TheFourier transform of the Riesz potential is understoodin the weak sense, considering the
space (Rd). The following lemma provides such a transform (see Lemma 1 of [41], p.117):
S
Lemma 2. Let us consider 0 < β < d.
(i) The Fourier transform of the function z d+β is γ(β) z β, in the sense that
− −
k k k k
(3.4) z d+βψ(z)dz = γ(β) z β (ψ)(z)dz, ψ (Rd),
− −
Rdk k Rd k k F ∀ ∈ S
Z Z
where
(ψ)(z) = exp( i x,z )ψ(x)dx
F Rd − h i
Z
denotes the Fourier transform of ψ.
(ii) The identity ( ∆) β/2(f) (z) = z β (f)(z) holds in the sense that
− −
F − k k F
(cid:0) (cid:1) 1
(3.5) ( ∆) β/2(f)(x)g(x)dx = (f)(x) x β (g)(x)dx, f,g (Rd).
− −
Rd − (2π)d RdF k k F ∀ ∈ S
Z Z
In particular, the following convolution formula is obtained by iteration of (3.5) using (3.2):
1 1
x y d+β y z d+βf(z)dz dy g(x)dx
− −
ZRd(cid:18)γ(β)ZRdk − k (cid:20)γ(β)ZRdk − k (cid:21) (cid:19)
= ( ∆) β/2 ( ∆) β/2(f) (x) g(x)dx
− −
Rd − −
Z h i
1
= (( ∆) β/2(f))(x) x β (g)(x)dx
− −
(2π)d Rd F − k k F
Z h i
NON-CENTRAL LIMIT THEOREMS 7
1
= (f)(x) x β x β (g)(x)dx
− −
(2π)d RdF k k k k F
Z
1
= (f)(x) x 2β (g)(x)dx
−
(2π)d RdF k k F
Z
= ( ∆) β(f)(x)g(x)dx, f,g (Rd), 0 < β < d/2,
−
Rd − ∀ ∈S
Z
(3.6)
where we have used that if f (Rd), then ( ∆) β/2(f) (Rd). From equation (3.6), and
−
∈ S − ∈ S
Lemma 2(i),
1
z d+2βf(z)dz = z 2β (f)(z)dz
− −
Rd γ(2β)k k Rdk k F
Z Z
1
= z y d+β y d+βdy f(z)dz, f (Rd), 0 <β < d/2.
− −
ZRd [γ(β)]2 (cid:20)ZRdk − k k k (cid:21) ∀ ∈ S
(3.7)
Let us now consider on the space of infinitely differentiable functions with compact support
contained in D, C (D) (Rd), the norm
0∞ ⊂ S
f 2 = ( ∆)α d/2(f),f = ( ∆)α d/2(f),f
k k(−∆)α−d/2 − − L2(Rd) − − L2(D)
D E D E
1 1
= ( ∆)α d/2(f)(x)f(x)dx = f(y)f(x)dydx
−
Rd − Rd γ(d 2α) Rd x y 2α
Z Z − Z k − k
1
(3.8) = (f)(λ)2 λ (d 2α)dλ, f C (D), 0< α < d/2.
(2π)d Rd|F | k k− − ∀ ∈ 0∞
Z
The associated inner product is given by
1 1
f,g = f(y)g(x)dydx
h i(−∆)α−d/2 Rd γ(d 2α) Rd x y 2α
Z − Z k − k
1 1
(3.9) = f(y)g(x)dydx,
γ(d 2α) x y 2α
ZD − ZD k − k
for all f,g ∈ C0∞(D). The closure of C0∞(D) with the norm k·k( ∆)α−d/2, introduced in (3.8),
−
defines a Hilbert space, which will be denoted as = C (D)k·k(−∆)α−d/2.
H2α−d 0∞
Remark 2. For a bounded open domain D, from Proposition 2.2. in [9], with D = n 1,
−
p = q = 2, and s = 0 (hence, As (D) = A0 (D) = L2(D), where, as usual, L2(D) denotes the
pq 22
space of square integrable functions on D), we have
(3.10) C (D)k·kL2(Rd) = L2(D),
0∞
(see also [45], for the case of regular bounded open domains with C boundaries). In addition,
∞
−
for all f C (D), by definition of the norm (3.8),
∈ 0∞
kfk( ∆)α−d/2 ≤ CkfkL2(Rd),
−
that is, all convergent sequences of C (D) in the L2(Rd) norm are also convergent in the
0∞
norm. Hence, the closure of C (D), with respect to the norm , is included in
H2α−d 0∞ k·kL2(Rd)
the closure of C0∞(D), with respect to the norm k·k( ∆)α−d/2. Therefore, from equation (3.10),
−
(3.11) L2(D) = C (D)k·kL2(Rd) C (D)k·k(−∆)α−d/2 = .
0∞ ⊆ 0∞ H2α−d
8 N.N.LEONENKO,M.D.RUIZ-MEDINA,ANDM.S.TAQQU
The asymptotic order of the eigenvalues of operator on L2(D), in the case d 2, are
α
K ≥
given in the next result (see, for example, [47], [49] and [51], p.197). (See also [13] and [48], for
the case d = 1).
Theorem1. Letusconsider theintegraloperator introduced inequation(3.1)asanoperator
α
on the space L2(D), with D being a bounded openKdomain of Rd. The following asymptotic is
satisfied by the eigenvalues λ ( ), k 1, of operator :
k α α
K ≥ K
λ ( )
(3.12) lim k Kα = c(d,α)D (d−α)/d,
k k (d α)/d | |
−→∞ − −
where D denotes, as before, the Lebesgue measure of domain D, and
| | e
(3.13) c(d,α) = πα/2 2 (d−α)/d Γ d−2α .
(cid:18)d(cid:19) Γ α Γ(cid:0) d (cid:1)(d−α)/d
2 2
Proof. We apply the reesults derived in [49], on th(cid:0)e a(cid:1)s(cid:2)ym(cid:0)pt(cid:1)o(cid:3)tic behavior of the eigenvalues
associated with certain class of integral equations. Specifically, the following integral equation
is considered in that paper:
(3.14) V1/2(x)k(x y)V1/2(y)f(y)dy = λf(x),
−
Z
where k is an integrable function over a Euclidean space E of dimension d, having positive
d
Fourier transform, and where V is a bounded non-negative function with bounded support. In
particular, [49] considers the case where E = Rd, V is the indicator function of a bounded
d
domain D Rd, and k( x y ) = x y α, for α > d, and α = 0,2,4,.... Function k
coincides in⊆Rd D with ak fu−nctikon whkose−Fokurier transfor−m f(ξ) is a6symptotically equal to
\
Γ d α
2d απd/2 −2 ξ d+α
− Γ α | |−
(cid:0) 2 (cid:1)
(see also the right-hand side of equation (3.4) for β = d α, with 0 < α < d). For α >
(cid:0) (cid:1)
−
d, α = 0,2,4,..., the following asymptotic of the eigenvalues of the integral operator with
− 6
kernel k( x y ) = x y α is given in equation (2) in [49]:
k − k k − k
2 d+dα Γ d+α (d+α)/d
(3.15) λ π α/2 2 [V(x)]d/(d+α)dx k (d+α)/d,
k − −
∼ (cid:18)d(cid:19) Γ −2α Γ(cid:0) d2 (cid:1)(d+α)/d (cid:20)ZRd (cid:21)
with
(cid:0) (cid:1)(cid:2) (cid:0) (cid:1)(cid:3)
[V(x)]d/(d−α)dx= D .
Rd | |
Z
Since function k in [49] coincides with the kernel of the integral operator in equation (3.1),
α
K
for α ( d,0), equation (3.15) then leads to the following asymptotic of the eigenvalues of :
α
∈ − K
λk( α) πα/2 2 d−dα Γ d−2α [V(x)]d/(d−α)dx (d−α)/dk−(d−α)/d.
K ∼ (cid:18)d(cid:19) Γ α2 Γ(cid:0) d2 (cid:1)(d−α)/d (cid:20)ZRd (cid:21)
(cid:3)
(cid:0) (cid:1)(cid:2) (cid:0) (cid:1)(cid:3)
Remark 3. Similar results to those ones presented in Theorem 3.2 of [48] can be derived for the
spectral zeta function of the Dirichlet Laplacian on a bounded closed multidimensional interval
of Rd (see also [13], for the case of d= 1). For a continuous function of the negative Dirichlet
Laplacian, the explicit computation of its trace cannot always be obtained in a general regular
bounded open domain of Rd. Specifically, the knowledge of the eigenvalues is guaranteed for
highly symmetric regions like the the sphere, or regions bounded by parallel planes (see, for
example, [30]; [31]; [32]). In particular, for the torus T2 in R2, the Spectral Zeta Function can
be explicitly computed (see, for example, [6], Chapter 1, equation (1.49), pp. 28-29).
NON-CENTRAL LIMIT THEOREMS 9
Thefollowingconditionisassumedtobesatisfiedbytheslowlyvaryingfunction inTheorem
L
2 below.
Condition A2. For every m 2, there exists a constant C > 0 such that
≥
(T x x ) (T x x ) (T x x )
1 2 2 3 m 1
..(m). L k − k L k − k L k − k dx dx dx
(T) x x α (T) x x α ··· (T) x x α 1 2 ··· m ≤
ZD ZD L k 1 − 2k L k 2 − 3k L k m− 1k
dx dx dx
1 2 m
(3.16) C ...(m). ··· .
≤ x x α x x α x x α
ZD ZD k 1− 2k k 2 − 3k ···k m− 1k
Note that Condition A2 is satisfied by slowly varying functions such that
(T x x )
1 2
(3.17) sup L k − k C ,
0
(T) ≤
T,x1,x2∈D L
for 0 < C 1. This condition holds for bounded slowly varying functions as in (1.6), in the
0
case where D≤ (0), with (0) = x Rd, x 1 .
1 1
⊆ B B { ∈ k k ≤ }
For the derivation of the limit distribution, when T , of the functional (1.1), we first
−→ ∞
compute its variance, in terms of H , the Hermite polynomial of order 2. It is well-known that
2
Hermite polynomials form a complete orthogonal system of the Hilbert space L (R,ϕ(u)du),
2
the space of square integrable functions with respect to the standard normal density ϕ. They
are defined as follows:
Hk(u) = (−1)keu22 ddukke−u22, k = 0,1,....
In particular, for a zero-mean Gaussian random field Y, for k 1,
≥
(3.18) E H (Y(x)) = 0, E (H (Y(x)) H (Y(y))) = δ m! (E[Y(x)Y(y)])m
k k m m,k
(see, for example, [33]).
We use some ideas from [17], Sections 1.4, 1.5 and 2.1). Consider the uniform distribution
on D(T) with the density:
(3.19) P (x) = T d D 1I , x Rd,
D(T) − − x D(T)
| | ∈ ∈
where I denotes the indicator function of set D(T).
x D(T)
Let U∈and V be two independent and uniformly distributed inside the set D(T) random
vectors. We denote ψ (ρ), the density of the Euclidean distance U V . Note that
D(T) k − k
ψ (ρ) = 0, if ρ > diam(D(T)), and ψ (ρ) is bounded, where diam(D(T)) is the di-
D(T) D(1)
ameter of the set D(T).
Using the above notation, we obtain
G( x y )dxdy = D(T)2E[G( U V )]
k − k | | k − k
ZD(T)ZD(T)
diam(D(T))
(3.20) = D 2T2d G(ρ)ψ (ρ)dρ,
| | D(T)
Z0
for any Borel function G such that the Lebesgue integral (3.20) exists. In particular, under
Conditions A1–A2 for 0 < α < d/2, and T , we obtain
→ ∞
2( x y )
σ2(T) = Var H (Y(x))dx = 2 L k − k dxdy
2 x y 2α
"ZD(T) # ZD(T)ZD(T) k − k
diam(D(T))
(3.21) = 2!D 2T2d 2(ρ)ρ 2αψ (ρ)dρ.
| | L − D(T)
Z0
10 N.N.LEONENKO,M.D.RUIZ-MEDINA,ANDM.S.TAQQU
In equation (3.21), consider the change of variable u = ρ/T. Applying the consistency of the
uniformdistribution with a homothetic transformation, and the asymptotic properties of slowly
varying functions (see Theorem 2.7 in [38]) we get
diam(D)
σ2(T) = 2D 2T2d 2α u 2α 2(uT)ψ (u)du
− − D
| | L
Z0
(3.22) = D 2T2d 2α 2(T)[a (D)]2(1+o(1)), 0 < α< d/2, T ,
− d
| | L → ∞
where, by (3.20),
diam(D) 1/2 dxdy 1/2
(3.23) a (D) = 2 u 2αψ (u)du = 2 .
d − D x y 2α
" Z0 # (cid:20) ZDZD k − k (cid:21)
More details, including properties of slowly varying functions, can be found in [3].
If D is the ball (0) = x Rd : x T , then (see [17], Lemma 1.4.2)
T
B { ∈ k k ≤ }
d+1 1
(3.24) ψ (ρ) = T dI , dρ, 0 ρ 2T,
BT(0) − 1−(2ρT)2(cid:18) 2 2(cid:19) ≤ ≤
where
Γ(p+q) µ
(3.25) I (p,q) = tp 1(1 t)q 1dt, µ [0,1], p > 0, q > 0,
µ − −
Γ(p)Γ(q) − ∈
Z0
is the incomplete beta function. In this case, one can show (see Lemma 2.1.3 in [17])
2d 2α+2πd 1/2Γ d 2α+1
(3.26) a (B (0)) = − − −2 .
d 1 (d 2α)Γ d Γ(d α+1)
− 2 (cid:0)− (cid:1)
For d= 1, D = [0,1],
(cid:0) (cid:1)
1 1 dxdy 1
a ([0,1]) = 2 = , 0 < α < 1/2.
1 x y 2α (1 α)(1 2α)
Z0 Z0 | − | − −
Let us now consider domain D = [0,l ] [0,l ] Rd, l > 0, i = 1,...,d. The Dirich-
1 d i
×···× ⊂
let negative Laplacian operator on such a domain has eigenvectors φ and eigenvalues
{ k}k≥1
λ ( ∆) given by
k k 1
{ − } ≥
d
πk x
i i
φ (x) = sin , x = (x ,...,x ) [0,l ] [0,l ], k 1, i = 1,...,d
k l 1 d ∈ 1 ×···× d i ≥
i=1 (cid:18) i (cid:19)
Y
d π2k2
λ ( ∆) = i , k= (k ,...,k ), k 1, i = 1,...,d.
k − l2 1 d i ≥
i=1 i
X
(3.27)
The norm 2 of 2 in (L2(D)), the space of bounded linear operators on L2(D),
kKαkL(L2(D)) Kα L
is given by the supremum of its eigenvalues. From Theorem 4.5(ii) in [10],
α d
d π2k2 −
(3.28) λ ( 2) 2 i , d α (0,1).
k Kα ≤ l2 − ∈
"i=1 i #
X
Note that Theorem 4.5(ii) in [10] holds for a bounded convex domain in Rd. From equation
(3.28), if min l ,...,l dπ,
{ 1 d} ≤ 2
(3.29) q 2 1.
kKαkL(L2(D)) ≤
Theorem 2. Let D be a regular bounded open domain. Assume that Conditions A1 and A2
are satisfied. The following assertions then hold: