Functional Limit Theorems for Toeplitz Quadratic Functionals of Continuous time Gaussian Stationary Processes

0.2 MB·English

by  Shuyang Bai

Functional 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