Table Of ContentFunctional Limit Theorems for Toeplitz Quadratic Functionals of
Continuous time Gaussian Stationary Processes
Shuyang Bai, Mamikon S. Ginovyan, Murad S. Taqqu
5 Boston University
1
0
April 30, 2015
2
r
p
A Abstract
The paper establishes weak convergence in C[0,1] of normalized stochastic processes, generated by
9
2 Toeplitz type quadratic functionals of a continuous time Gaussian stationary process, exhibiting long-
range dependence. Both central and non-centralfunctional limit theorems are obtained.
]
R
Key words. Stationary Gaussian process - Toeplitz-type quadratic functional - Brownian motion - Non-
P
central limit theorem - Long memory - Wiener-Itoˆ integral.
.
h
t
a 1 Introduction
m
[ Let X(t), t R be a centered real-valued stationary Gaussian process with spectral density f(x) and
{ ∈ }
2 covariance function r(t), that is, r(t) = f(t) = eixtf(x)dx, t R. We are interested in describing the
R
v ∈
limit (as T ) of the following process, generated by Toeplitz type quadratic functionals of the process
4 → ∞ R
X(t): b
7
Tt Tt
5
Q (t)= g(u v)X(u)X(v)dudv, t [0,1], (1.1)
5 T − ∈
0 Z0 Z0
. where b
1
0 g(t)= eixtg(x)dx, t R, (1.2)
R ∈
5 Z
1 istheFouriertransformofsomeintegrableevenfunctiong(x), x R. Wewillrefertog(x)andtoitsFourier
b
v: transform g(t) as a generating function and generating kernel fo∈r the process QT(t), respectively.
i Thelimitoftheprocess(1.1)iscompletelydeterminedbythespectraldensityf(x)(orcovariancefunction
X
r(t)) and the generating function g(x) (or generating kernel g(t)), and depending on their properties, the
b
r
limitcanbeeitherGaussian(thatis,Q (t)withanappropriatenormalizationobeysacentrallimittheorem),
a T
or non-Gaussian. The following two questions arise naturally:
b
(a) Under what conditions on f(x) (resp. r(t)) and g(x) (resp. g(t)) will the limit be Gaussian?
(b) Describe the limit process, if it is non-Gaussian.
b
Similar questions were considered by Fox and Taqqu [6], Ginovyan and Sahakyan [9], and Terrin and
Taqqu [25] in the discrete time case.
Here we work in continuous time, and establish weak convergence in C[0,1] of the process (1.1). The
limit processes can be Gaussian or non-Gaussian. The limit non-Gaussian process is identical to that of in
the discrete time case, obtained in [25].
But first some brief history. The question (a) goes back to the classical monograph by Grenander and
Szego¨ [16], where the problem was considered for discrete time processes, as an application of the authors’
1
theory of the asymptotic behavior of the trace of products of truncated Toeplitz matrices (see [16], p.
217-219). Later the question (a) was studied by Ibragimov [17] and Rosenblatt [23], in connection to the
statistical estimation of the spectral function F(x) and covariance function r(t), respectively. Since 1986,
there has been a renewedinterest in both questions (a) and (b), related to the statistical inferences for long
memoryprocesses(see,e.g.,Avram[1],FoxandTaqqu[6],GinovyanandSahakyan[9],Ginovyanetal. [11],
Giraitisetal. [12],GiraitisandSurgailis[13],Giraitis andTaqqu[15], TerrinandTaqqu[26],Taniguchiand
Kakizawa [24], and references therein). In particular, Avram [1], Fox and Taqqu [6], Giraitis and Surgailis
[13], Ginovyan and Sahakyan [9] have obtained sufficient conditions for the Toeplitz type quadratic forms
Q (1) to obey the central limit theorem (CLT), when the model X(t) is a discrete time process.
T
For continuous time processes the question (a) was studied in Ibragimov [17] (in connection to the
statistical estimation of the spectral function), Ginovyan [7, 8], Ginovyan and Sahakyan [10] and Ginovyan
etal. [11],wheresufficientconditionsintermsoff(x)andg(x)ensuringcentrallimittheoremsforquadratic
functionals Q (1) have been obtained.
T
The rest of the paper is organized as follows. In Section 2 we state the main results of this paper
(Theorems 2.1 - 2.4). In Section 3 we prove a number of preliminary lemmas that are used in the proofs of
the main results. Section 4 contains the proofs of the main results.
Throughout the paper the letters C and c with or without indices will denote positive constants whose
values can change from line to line.
2 The Main Results
In this section we state our main results. Throughout the paper we assume that f,g L1(R), and with no
∈
loss of generality, that g 0 (see [10], [13]).
≥
We first examine the case of central limit theorems, and consider the following standard normalized
version of (1.1):
Q (t):=T−1/2(Q (t) IE[Q (t)]), t [0,1]. (2.1)
T T T
− ∈
Ourfirstresult,whichisanextensionofTheorem1of[10],involvestheconvergenceoffinite-dimensional
e
distributions of the process Q (t) to that of a standard Brownian motion.
T
Theorem 2.1. Assume that the spectral density f(x) and the generating function g(x) satisfy the following
e
conditions:
f g L1(R) L2(R) (2.2)
· ∈ ∩
and
∞
IE[Q2(1)] 16π3 f2(x)g2(x)dx as T . (2.3)
T → ∞ →∞
Z
Then we have the following conveergence of finite-dimensional distributions
f.d.d.
Q (t) σB(t),
T
−→
where Q (t) is as in (2.1), B(t) is a standard Brownian motion, and
T e
∞
e σ2 :=16π3 f2(x)g2(x)dx. (2.4)
−∞
Z
To extendthe convergenceof finite-dimensionaldistributions in Theorem2.1to the weak convergencein
the space C[0,1], we impose an additional condition on the underlying Gaussian process X(t) and on the
generatingfunctiong. Itisconvenienttoimposethisconditioninthetimedomain,thatis,onthecovariance
function r :=fˆand the generating kernel a:=gˆ. The following condition is an analog of the assumption in
Theorem 2.3 of [15]:
1 1 3
r() Lp(R), a() Lq(R) for some p,q 1, + . (2.5)
· ∈ · ∈ ≥ p q ≥ 2
2
Remark 2.1. In fact under (2.2), the condition (2.5) is sufficient for the convergence in (2.3). Indeed, let
p¯ = p/(p 1) be the Ho¨lder conjugate of p and let q¯ = q/(q 1) be the Ho¨lder conjugate of q. Since
− −
1 p,q 2, one has by the Hausdorff-Young inequality and (2.5) that f c r , g c a , and
p¯ p p q¯ q q
≤ ≤ k k ≤ k k k k ≤ k k
hence
1 1 1 1
f() Lp¯, g() Lq¯, + =2 1/2.
· ∈ · ∈ p¯ q¯ − p − q ≤
Then the convergence in (2.3) follows from the proof of Theorem 3 from [10]. Note that a similar assertion
in the discrete time case was established in [13].
Remark 2.2. Observe that condition (2.5) is fulfilled if the functions r(t) and a(t) satisfy the following:
∗ ∗
there exist constants C >0, α and β , such that
r(t) C(1 tα∗−1), a(t) C(1 tβ∗−1), (2.6)
| |≤ ∧| | | |≤ ∧| |
where 0 < α∗,β∗ < 1/2 and α∗ +β∗ < 1/2. Indeed, to see this, note first that r(), a() L∞(R). Then
one can choose p,q 1 such that p(α∗ 1)< 1 and q(β∗ 1)< 1, which entai·ls tha·t r∈() Lp(R) and
a() Lq(R). Since≥1/p+1/q < 2 α∗− β∗ a−nd 2 α∗ −β∗ > 3/−2, one can further choose· p∈,q to satisfy
· ∈ − − − −
1/p+1/q 3/2.
≥
The nextresults,twofunctional centrallimittheorems,extend Theorems1and5 of[10]to weakconver-
gence in the space C[0,1] of the stochastic process Q (t) to a standard Brownian motion.
T
Theorem 2.2. Letthespectraldensity f(x)and thegeneratingfunction g(x)satisfy condition (2.2). Let the
e
covariance function r(t) and the generating kernel a(t) satisfy condition (2.5). Then we have the following
weak convergence in C[0,1]:
Q (t) σB(t),
T
⇒
where Q (t) is as in (2.1), σ is as in (2.4), and B(t) is a standard Brownian motion.
T e
Receall that a function u(x), x R, is called slowly varying at 0 if it is non-negative and for any t>0
∈
u(xt)
lim 1.
x→0 u(x) →
Let SV (R) be the class of slowly varying at zero functions u(x), x R, satisfying the following conditions:
0
∈
for some a > 0, u(x) is bounded on [ a,a], limx→0u(x) = 0, u(x) = u( x) and 0 < u(x) < u(y) for
0<x<y <a. An example of a functio−n belonging to SV (R) is u(x)= ln x−−γ with γ >0 and a=1.
0
| | ||
Theorem 2.3. Assumethatthefunctionsf andg areintegrableon Randboundedoutsideanyneighborhood
of the origin, and satisfy for some a>0
f(x) x−αL (x), g(x) x−βL (x), x [ a,a] (2.7)
1 2
≤| | | |≤| | ∈ −
for some α < 1, β < 1 with α+β 1/2, where L (x) and L (x) are slowly varying at zero functions
1 2
≤
satisfying
L SV (R), x−(α+β)L (x) L2[ a,a], i=1,2. (2.8)
i 0 i
∈ ∈ −
Let, in addition, the covariance function r(t) and the generating kernel a(t) satisfy condition (2.5). Then
we have the following weak convergence in C[0,1]:
Q (t) σB(t),
T
⇒
where QT(t) is as in (2.1), σ is as in (2.4), aned B(t) is a standard Brownian motion.
Remark 2.3. The conditions α < 1 and β < 1 ensure that the Fourier transforms of f and g are well
e
defined. Observe that when α > 0 the process X(t),t Z may exhibit long-range dependence. We also
{ ∈ }
allow here α+β to assume the critical value 1/2.
3
Remark 2.4. The assumptions f g L1(R), f,g L∞(R [ a,a]) and (2.8) imply that f g L2(R), so
· ∈ ∈ \ − · ∈
that σ2 in (2.4) is finite.
Remark 2.5. One may wonder, why, in Theorem 2.3, we suppose that L (x) and L (x) belong to SV (R)
1 2 0
insteadofmerelybeingslowlyvaryingatzero. Thisisdoneinordertodealwiththecriticalcaseα+β =1/2.
Suppose that we are away from this critical case, namely, f(x) = x−αl (x) and g(x) = x−βl (x), where
1 2
| | | |
α+β < 1/2, and l (x) and l (x) are slowly varying at zero functions. Assume also that f(x) and g(x) are
1 2
integrable and bounded on ( , a) (a,+ ) for any a>0. We claim that Theorem 2.3 applies. Indeed,
choose α′ > α, β′ > β with−α∞′ +−β′ ∪< 1/2.∞Write f(x) = x−α′ xδl (x), where δ = α′ α > 0. Since
1
l (x) is slowly varying, when x is small enough, for some ǫ| |(0,δ|)|we have xδl (x) x−δ−ǫ. Then one
1 1
can bound xδ−ǫ by c ln x −|1| SV (R) for small x < 1. ∈Hence one has w|he|n x <≤1|is| small enough,
0
| | | | || ∈ | | | |
f(x) x−α′ c ln x −1 . Similarly, when x <1 is smallenough,one has g(x) x−β′ c ln x −1 . All
≤| | | | || | | ≤| | | | ||
′ ′
the assumptio(cid:16)ns in Theor(cid:17)em 2.3 are now readily checked with α, β replaced by α and β ,(cid:16)respectively(cid:17).
Nowwestate anon-central limit theorem inthe continuoustimecase. Letthe spectraldensityf andthe
generating function g satisfy
f(x)= x−αL (x) and g(x)= x−βL (x), x R, α<1, β <1, (2.9)
1 2
| | | | ∈
withslowlyvaryingatzerofunctionsL (x)andL (x)suchthat x−αL (x)dx< and x−βL (x)dx<
1 2 R| | 1 ∞ R| | 2
. WeassumeinadditionthatthefunctionsL (x)andL (x)satisfythefollowingcondition,calledPotter’s
1 2
∞ R R
bound (see [12], formula (2.3.5)): for any ǫ > 0 there exists a constant C > 0 so that if T is large enough,
then
L (u/T)
i C(uǫ+ u−ǫ), i=1,2. (2.10)
L (1/T) ≤ | | | |
i
Note that a sufficient condition for (2.10) to hold is that L (x) and L (x) are bounded on intervals [a, )
1 2
∞
for any a>0, which is the case for the slowly varying functions in Theorem 2.3.
Now we areinterested inthe limit processof the followingnormalizedversionofthe process Q (t) given
T
by (1.1), with f and g as in (2.9):
1
Z (t):= (Q (t) IE[Q (t)]). (2.11)
T Tα+βL (1/T)L (1/T) T − T
1 2
Theorem 2.4. Let f and g be as in (2.9) with α<1, β <1 and slowly varying at zero functions L (x) and
1
L (x) satisfying (2.10), and let Z (t) be as in (2.11). Then for α+β > 1/2, we have the following weak
2 T
convergence in the space C[0,1]:
Z (t) Z(t),
T
⇒
where the limit process Z(t) is given by
′′
Z(t)= H (x ,x )W(dx )W(dx ), (2.12)
t 1 2 1 2
R2
Z
with
eit(x1+u) 1 eit(x2−u) 1
H (x ,x )= x x −α/2 − − u−βdu , (2.13)
t 1 2 1 2
| | ZR(cid:20) i(x1+u) (cid:21)·(cid:20) i(x2−u) (cid:21)| |
where W() is a complex Gaussian random measure with Lebesgue control measure, and the double prime in
·
the integral (2.12) indicates that the integration excludes the diagonals x = x .
1 2
±
Remark 2.6. ComparingTheorem2.4andTheorem1of[25],weseethatthelimitprocessZ(t)isthesame
both for continuous and discrete time models.
Remark 2.7. Denoting by P and P the measures generated in C[0,1] by the processes Z (t) and Z(t)
T T
given by (2.11) and (2.12), respectively, Theorem 2.4 can be restated as follows: under the conditions of
Theorem 2.4, the measure P converges weakly in C[0,1] to the measure P as T . A similar assertion
T
→∞
can be stated for Theorems 2.2 and 2.3.
4
ItisworthnotingthatalthoughthestatementofourTheorem2.4issimilartothatofTheorem1of[25],
the proofis differentandsimpler,anddoesnotusethe hardanalysisof[25], althoughsometechnicalresults
of [25] are stated in lemmas and used in the proofs. Our approach in the CLT case (Theorems 2.1 - 2.3),
uses the method developed in [10], which itself is based on an approximation of the trace of the product of
truncated Toeplitz operators. For the non-CLT case (Theorem 2.4), we use the integral representation of
the underlying process and properties of Wiener-Itoˆ integrals.
3 Preliminaries
In this section we state a number of lemmas which will be used in the proof of the theorems. The following
result extends Lemma 9 of [10].
Lemma3.1. LetY(t)beacenteredstationaryGaussianprocesswithspectraldensityf (x) L1(R) L2(R).
Y
∈ ∩
Consider the normalized process:
1 Tt Tt
L (t):= Y2(u)du IE Y2(u)du . (3.1)
T T1/2 Z0 − "Z0 #!
Then we have the following convergence of finite-dimensional distributions:
∞
L (t)f.d.d.σ B(t), σ2 =4π f2(x)dx, (3.2)
T −→ Y Y −∞ Y
Z
where B(t) is standard Brownian motion.
Remark 3.1. Observe that the normalized processes Q (t) and L (t), given by (2.1) and (3.1), can be
T T
expressed by double Wiener-Itoˆ integrals (see, e.g., the proof of Lemma 3.9 below). In our proofs we will
use the followingfactaboutweakconvergenceofmultipleeWiener-Itoˆ integrals: giventhe convergenceofthe
covariance, the multivariate convergence to a Gaussian vector is implied by the univariate convergence of
each component (see [21], Proposition2).
Proof of Lemma 3.1. For a fixed t, the univariate convergence in distribution
L (t) d N(0,tσ2) as T
T → Y →∞
follows from Lemma 9 of [10]. To show (3.2), in view of Remark 3.1 and Proposition 2 of [21], it remains
to show that the covariance structure of L (t) converges to that of σ B(t). Specifically, it suffices to show
T Y
that for any 0<s<t,
IE (L (t) L (s))2 σ2 (t s) as T . (3.3)
T − T → Y · − →∞
h i
Indeed, using the fact that for a Gaussian vector (G ,G ) we have Cov(G2,G2) = 2[Cov(G ,G )]2, and
1 2 1 2 1 2
letting r (u)= eixuf (x)dx be the covariance function of Y(t), we can write
Y R Y
R T(t−s) u
IE (L (t) L (s))2 =2(t s) 1 | | r2(u)du.
h T − T i − Z−T(t−s)(cid:18) − T(t−s)(cid:19) Y
Since f (x) L2(R), the Fourier transform r (u) L2(R) as well. So by the Dominated Convergence
Y Y
∈ ∈
Theorem and Parseval-Plancherel’sidentity, we have as T
→∞
∞ ∞
IE (L (t) L (s))2 2(t s) r2(u)du=4π(t s) f2(x)dx=σ2(t s). (3.4)
T − T → − −∞ Y − −∞ Y Y −
h i Z Z
5
We now discuss some results which allow one to reduce the general quadratic functional in Theorem 2.1
to a special quadratic functional introduced in Lemma 3.1.
By Theorem 16.7.2 from [18], the underlying process X(t) admits a moving average representation:
∞ ∞
X(t)= aˆ(t s)B(ds) with aˆ(t)2dt< , (3.5)
−∞ − −∞| | ∞
Z Z
where B(t) is a standard Brownian motion, and aˆ(t) is such that its inverse Fourier transform a(x) satisfies
f(x)=2π a(x)2. Assuming the conditions (2.2) and (2.3), we set
| |
b(x)=(2π)1/2a(x)(g(x))1/2,
and observe that the function b(x) is then in L2(R) due to condition (2.2). Consider the stationary process
∞
Y(t)= ˆb(t s)B(ds) (3.6)
−∞ −
Z
constructed using the Fourier transform ˆb(t) of b(x) and the same Brownian motion B(t) as in (3.5). The
process Y(t) has spectral density (see [10], equation (4.7))
f (x)=2πf(x)g(x). (3.7)
Y
We have the following approximation result which immediately follows from Lemma 10 of [10].
Lemma 3.2. Let Q (t) be as in (2.1) and let L (t) be as in (3.1) with Y(t) constructed as in (3.6). Then
T T
under the conditions (2.2) and (2.3), for any t>0, we have
e
lim Var[Q (t) L (t)]=0.
T T
T→∞ −
The following lemma is a straightforwardadapetation of Lemma 4.2 of [14] for functions defined on R.
Lemma 3.3. If p 1, j =1,...,k, where k 2 and k 1 =k 1, then
j ≥ ≥ j=1 pj −
P
k
ZRk−1|f1(x1)...fk−1(xk−1)fk(x1+...+xk−1)|dx1...dxk ≤j=1kfjkpj.
Y
The following lemma will be used to establish tightness in the space C[0,1] in Theorem 2.2.
Lemma 3.4. Let the covariance function r(t) and the generating kernel a(t) satisfy condition (2.5), and let
Q (t) be as in (2.1). Then for all 0 s t 1 and T >0, there exists a constant C >0, such that
T
≤ ≤ ≤
e IE Q (t) Q (s)2 C(t s). (3.8)
T T
| − | ≤ −
h i
Proof. For convenience we use the Wick eproduct enotation: : X(u)X(v) := X(u)X(v) IE[X(u)X(v)]. So
−
for 0 s t 1, we can write
≤ ≤ ≤
1 Tt Tt Ts Ts
Q (t) Q (s)= a(u v):X(u)X(v):dudv a(u v):X(u)X(v):dudv
T T
− √T Z0 Z0 − −Z0 Z0 − !
e e 1 Tt Tt 2 Ts Tt
= a(u v):X(u)X(v):dudv+ a(u v):X(u)X(v):dudv
√T − √T −
ZTs ZTs Z0 ZTs
:=A(s,t,T)+B(s,t,T).
6
Now we estimate B(s,t,T) (the function A(s,t,T) can be estimated similarly). We have by Theorem 3.9 of
[19] that
4 Ts Tt Ts Tt
IE B2(s,t,T) = du dv du dv a(u v )a(u v )IE(:X(u )X(v )::X(u )X(v ):)
1 1 2 2 1 1 2 2 1 1 2 2
T − −
Z0 ZTs Z0 ZTs
4 (cid:2) Ts (cid:3)Tt Ts Tt
= du dv du dv a(u v )a(u v )[r(u u )r(v v )+r(u v )r(v u )]
1 1 2 2 1 1 2 2 1 2 1 2 1 2 1 2
T − − − − − −
Z0 ZTs Z0 ZTs
:=B (s,t,T)+B (s,t,T).
1 2
By the change of variables x =u v , x =v u , x =u u , x =v , and noting that r() and a()
1 1 1 2 2 2 3 2 1 4 2
− − − · ·
are even functions, we have
4 Tt
B (s,t,T) dx a(x )a(x )r(x )r(x +x +x )dx dx dx .
1 4 1 2 3 1 2 3 1 2 3
≤ T ZTs ZR3| |
Since r(t) r(0), we have r() L∞(R). We also have r() Lp(R) by condition (2.5), where 1/p+1/q
| |≤ · ∈ · ∈ ≥
3/2. The Lp-interpolationtheoremstates that if a function is in Lp1 and Lp2 with 0<p p , then it
1 2
is in Lp′, p p′ p . By the Lp-interpolation theorem, one can choose p′ p such that≤r() ≤L∞p′(R) and
1 2
≤ ≤ ≥ · ∈
1 1 1 1 1 1 3
+ + + =3, that is, + = .
p′ p′ q q p′ q 2
Then by Lemma 3.3, one has B (s,t,T) 4 r 2 a 2(t s). Similarly, one can establish the bound
1 ≤ k kp′k kq −
B (s,t,T) C(t s), and hence B(s,t,T) C(t s). So (3.8) is proved.
2
≤ − ≤ −
The lemmas that follow will be used in the proof of Theorem 2.4.
Lemma 3.5. Define
t eitx 1
∆ (x)= eisxds= − , (3.9)
t
ix
Z0
Then for any δ (0,1), there exists a constant c>0 depending only on δ, such that
∈
∆ (x) ctδf (x), t [0,1], x R, (3.10)
t δ
| |≤ | | ∈ ∈
where
xδ−1 if x >1;
f (x)= | | | | (3.11)
δ
(1 if x 1.
| |≤
Proof. In view of (3.9), we have ∆ (x) t eisx ds = t. So under the constraint t [0,1], we have
| t | ≤ 0 | | ∈
∆ (x) t tδ. On the other hand, from Lemma 2 from [25], with some constant C > 0, we have
| t | ≤ ≤ R
eix 1 C xδ, δ (0,1). So
| − |≤ | | ∈
eitx 1
∆ (x) | − | C txδ x−1 =Ctδ xδ−1.
t
| |≤ x ≤ | | | | | |
| |
Combining this with (3.11), we obtain (3.10).
We quote Lemma 1 of [25] in a special case, convenient for our purposes.
Lemma 3.6. Let γ <1, γ +γ >1/2, and let δ be such that
i i i+1
γ +γ
i i+1
0 δ < ,
≤ 2
7
where i=1,...,4 (with γ =γ ). Then
5 1
f (y y )f (y y )f (y y )f (y y )y −γ1 y −γ2 y −γ3 y −γ4dy< ,
δ 1 2 δ 2 3 δ 3 4 δ 4 1 1 2 3 4
R4 − − − − | | | | | | | | ∞
Z
where f () is as in (3.11).
δ
·
Lemma 3.6 can be used to establish the following result.
Lemma 3.7. The function
H∗(x ,x ):= x α1/2 x α2/2 ∆ (x +u)∆ (x u) u−βdu (3.12)
t 1 2 | 1| | 2| R| t 1 t 2− || |
Z
is in L2(R2) for all (α ,α ,β) in the open region (α ,α ,β): α ,α ,β <1, α +β >1/2, i=1,2 .
1 2 1 2 1 2 i
{ }
Proof. It suffices focus on the case where t [0,1], otherwise a change of variable can reduce it to this case.
∈
We have by suitable change of variables and Lemma 3.5 that
H∗ 2 = ∆ (y y )∆ (y y )∆ (y y )∆ (y y ) y −α1 y −β y −α2 y −βdy
k tkL2(R2) R4 t 1− 2 t 2− 3 t 3− 4 t 4− 1 | 1| | 2| | 3| | 4|
Z
(cid:12) (cid:12)
C (cid:12) f (y y )f (y y )f (y y )f (y y )y(cid:12) −α1 y −β y −α2 y −βdy.
δ 1 2 δ 2 3 δ 3 4 δ 4 1 1 2 3 4
≤ R4 − − − − | | | | | | | |
Z
Then apply Lemma 3.6, noting that δ can be chosen arbitrarily small.
Lemma 3.8. Define the function
H∗ (x ,x )=A (x ,x )x x −α/2 ∆ (x +u)∆ (x u) u−βA (u) du, (3.13)
t,T 1 2 1,T 1 2 | 1 2| R | t 1 t 2− || | 2,T
Z
where
L (x /T)L (x /T) L (u/T)
1 1 1 2 2
A (x ,x )= , A (u)= . (3.14)
1,T 1 2 2,T
s L1(1/T) L1(1/T) L2(1/T)
Then for large enough T, we have H∗ (x ,x ) L2(R2).
t,T 1 2 ∈
Proof. By (2.10) and (3.14), for any ǫ>0 there exists C >0, such that for T large enough,
A (x ,x ) C(x ǫ+ x −ǫ)(x ǫ+ x −ǫ) (3.15)
1,T 1 2 1 1 2 2
| |≤ | | | | | | | |
and
A (u) C(uǫ+ u−ǫ). (3.16)
2,T
| |≤ | | | |
Hence, with some constant C >0,
H∗ (x ,x ) C ∆ (x +u)∆ (x u) u−β(uǫ+ u−ǫ)du x x −α/2(x ǫ+ x −ǫ)(x ǫ+ x −ǫ).
| t,T 1 2 |≤ R | t 1 t 2− || | | | | | | 1 2| | 1| | 1| | 2| | 2|
Z
(3.17)
Because by Lemma 3.7, the function H∗ in (3.12) is in L2(R2) for all (α ,α ,β) in an open region (α,β):
t 1 2 {
α ,α ,β <1,α +β >1/2,i=1,2 . By choosingǫ smallenough, we infer that the right-handside of (3.17)
1 2 i
is in L2(R2), and the result follows}.
8
Lemma 3.9. Let Z (t) be as in (2.11), and let
T
′′
′
Z (t):= H (x ,x ) W(dx )W(dx ), (3.18)
T t,T 1 2 1 2
R2
Z
where
H (x ,x )=A (x ,x )x x −α/2 ∆ (x +u)∆ (x u)u−βA (u) du . (3.19)
t,T 1 2 1,T 1 2 1 2 t 1 t 2 2,T
| | R − | |
(cid:20)Z (cid:21)
f.d.d. ′ ′
ThenZ (t) = Z (t),thatis,theprocessesZ (t)andZ (t)havethesamefinite-dimensionaldistributions.
T T T T
Proof. UsingthespectralrepresentationofX(t)(see,e.g.,[5],ChapterXI,Section8): X(t)= eitx f(x)W(dx),
R
where W() is a complex Gaussian measure with Lebesgue control measure, and the diagram formula (see,
· R p
e.g., [20], Chapter 5), we have
′′
X(u)X(v) IE[X(u)X(v)]= ei(ux1+vx2) f(x )f(x )W(dx )W(dx ).
1 2 1 2
− R2
Z
p
By a stochastic Fubini Theorem (see [22], Theorem 2.1) and Lemma 3.8, one can change the integration
order to get (note that by (1.2) we have g(t)= eitxg(x)dx):
R
1 ′′ R Tt Tt
Z (t)= f(xb)f(x ) ei(u−v)wg(w)dw ei(ux1+vx2)dudv W(dx )W(dx )
T Tα+βL1(1/T)L2(1/T)ZR2 1 2 Z0 Z0 ZR 1 2
1 ′′p Tt Tt
= f(x )f(x ) eiu(x1+w)du eiv(x2−w)dv w−βL(w)dw W(dx )W(dx )
Tα+βL1(1/T)L2(1/T)ZR2 1 2 ZR Z0 Z0 | | 1 2
′′p
1
= f(x )f(x ) ∆ (x +w)∆ (x w)w−βL (w)dw W(dx )W(dx ).
Tα+βL1(1/T)L2(1/T)ZR2 1 2 ZR Tt 1 Tt 2− | | 2 1 2
p
Now we use the change of variables w u/T, x x /T, x x /T, where the latter two change of
1 1 2 2
→ → →
variables are subject to the rule W(dx/T)=d T−1/2W(dx) (see, e.g., [4], Proposition 4.2), to obtain
f.d.d. 1
Z (t) =
T Tα+βL (1/T)L (1/T)×
1 2
′′
f(x /T)f(x /T) ∆ (x +u)∆ (x u)w/T −βL (w/T)Tdw T−1W(dx )W(dx ).
1 2 t 1 t 2 2 1 2
R2 R − | |
Z Z
p (3.20)
Taking into account the equality f(x/T) = x/T −αL (x/T) and equations in (3.14), we see that the right
1
| |
hand side of (3.20) coincides with (3.18). This completes the proof.
The lemmas that follow will be used to establish tightness in the space C[0,1] in Theorem 2.4.
Lemma 3.10. Let δ be a fixed number within the range (0,(α+β)/2), and let Z (t) be as in (2.11). Then
T
for all 0 s t 1 and T large enough, there exists a constant C >0, such that
≤ ≤ ≤
IE Z (t) Z (s)2 C(t s)2δ. (3.21)
T T
| − | ≤ −
The same estimate also holds for the co(cid:2)rresponding limit(cid:3)ing process Z(t) defined by (2.12), (2.13).
Proof. First, in view of Lemma 3.9, we have IE Z (t) Z (s)2 = IE Z′ (t) Z′ (s)2 . Next, using the
| T − T | | T − T |
linearity of the multiple stochastic integral, we can write
(cid:2) (cid:3) (cid:2) (cid:3)
′′
′ ′
Z (t) Z (s)= H (x ,x )W(dx )W(dx ),
T − T R2 s,t,T 1 2 1 2
Z
9
where
H (x ,x )=A (x ,x )x x −α/2 [∆ (x +u)∆ (x u) ∆ (x +u)∆ (x u)] u−βA (u)du.
s,t,T 1 2 1,T 1 2 1 2 t 1 t 2 s 1 s 2 2,T
| | R − − − | |
Z
(3.22)
The term in the brackets of the integrand in (3.22) can be rewritten as follows:
∆ (x +u)∆ (x u) ∆ (x +u)∆ (x u)
t 1 t 2 s 1 s 2
− − −
t t s s
= eiw1(x1+u)eiw2(x2−u)dw dw eiw1(x1+u)eiw2(x2−u)dw dw
1 2 1 2
−
Z0 Z0 Z0 Z0
s t t s t t
= dw dw ...+ dw dw ...+ dw dw ...
1 2 1 2 1 2
Z0 Zs Zs Z0 Zs Zs
=∆s(x1+u)∆t−s(x2 u)+∆t−s(x1+u)∆s(x2 u)+∆t−s(x1+u)∆t−s(x2 u).
− − −
Now we apply Lemma 3.5 to get
∆ (x +u)∆ (x u) ∆ (x +u)∆ (x u) C[sδ(t s)δ+(t s)δsδ+(t s)2δ]f (x +u)f (x u)
t 1 t 2 s 1 s 2 δ 1 δ 2
| − − − |≤ − − − −
C(t s)δf (x +u)f (x u), (3.23)
δ 1 δ 2
≤ − −
where the last inequality follows because 0 sδ 1 and 0 (t s)δ 1.
′ ≤ ≤ ≤ − ≤
Next, using formula (4.5) of [20], (3.22) and (3.23), we can write
IE Z (t) Z (s)2 = H 2 C t s2δ dx dx A (x ,x )2 x x −α
| T − T | k s,t,TkL2(R2) ≤ | − | R2 1 2 1,T 1 2 | 1 2| ×
Z
(cid:2) (cid:3)
du du f (x +u )f (x u )f ( x +u )f ( x u )u −β u −βA (u )A (u )
1 2 δ 1 1 δ 2 1 δ 1 2 δ 2 2 1 2 2,T 1 2,T 2
R2 − − − − | | | |
Z
C t s2δ dy dy dy dy A (y ,y )2A (y )A (y )
1 2 3 4 1,T 1 3 2,T 2 2,T 4
≤ | − | R4 ×
Z
f (y y )f (y y )f (y y )f (y y )y −α y −β y −α y −β, (3.24)
δ 1 2 δ 2 3 δ 3 4 δ 4 1 1 2 3 4
− − − − | | | | | | | |
where we have applied the change of variables: y =x , y = u , y = x , y =u .
1 1 2 1 3 2 4 2
− −
Since by assumption α < 1, β < 1 and α+β > 1/2, and the exponent ǫ in (3.15) and (3.16) can be
chosenarbitrarilysmall,forafixedδ satisfying0<δ <(α+β)/2,wecanapplyLemma1of[25]toconclude
that the integral
dyA (y ,y )2A (y )A (y )f (y y )f (y y )f (y y )f (y y )y −α y −β y −α y −β
1,T 1 3 2,T 2 2,T 4 δ 1 2 δ 2 3 δ 3 4 δ 4 1 1 2 3 4
R4 − − − − | | | | | | | |
Z
is bounded for sufficiently large T, which in view of (3.24) implies (3.21). The proof for Z (t) is thus
T
complete. The proof for Z(t) is similar and so we omit the details.
4 Proof of Main Results
Proof of Theorem 2.1. By Lemma 3.2, for any 0 t <...<t , and constants c ,...,c , we have
1 n 1 n
≤
n
lim Var c Q (t ) L (t ) =0.
j T j T j
T→∞ −
Xj=1 (cid:16) (cid:17)
e
Therefore the convergence of finite-dimensional distributions of Q (t) to that of Brownian motion σB(t)
T
follows from Lemma 3.1 with f () given in (3.7) and the Cram´er-WoldDevice.
Y
·
e
10
