IDENTIFICACIÓN Y ESTIMACIÓN GENERAL DE RIESGO

Proof of Theorem 6.2.1. By Lemma 6.3.3, for any 0 ≤ t1 < . . . < tn, and constants

c1, . . . , cn, we have lim T →∞Var   n X j=1 cj  e QT(tj) − LT(tj)   = 0.

Therefore the convergence of finite-dimensional distributions of eQT(t) to that of Brownian

motion σB(t) follows from Lemma 6.3.1 with fY(·) given in (6.22) and the Cram´er-Wold

Device.

Proof of Theorem 6.2.4. In view of the well-known Prokhorov’s Theorem (see, e.g., Billings- ley [1999], p. 58), to prove the theorem, we need to show convergence of finite-dimensional distributions and tightness. The former has been established in Theorem 6.2.1. To prove tightness, observe that by Lemma 6.3.5 and the hypercontractivity inequality of the multi- ple Wiener-Itˆo integrals (see Major [2014], Corollary 5.6), for any T > 0 and 0 ≤ s ≤ t ≤ 1, there exists a constant C > 0 to satisfy

Eh| eQT(t) − eQT(s)|4

i

≤ C₂Eh| eQT(t) − eQT(s)|2

i2

≤ C(t − s)2_{. (6.40)}

Now the tightness of the family of measures generated by the processes { eQT(t) : T > 0} in

C[0, 1] follows from Lemma 5.1 of Ibragimov [1963].

Proof of Theorem 6.2.5. The convergence of finite-dimensional distributions follows from Theorem 6.2.1. In fact, the assumptions on f and g in Theorem 6.2.5 imply the conditions (6.4) and (6.5) in Theorem 6.2.1 (see the proof of Theorem 5 in Ginovyan and Sahakyan [2007]). The tightness can be shown similarly as in the proof of Theorem 6.2.4.

Proof of Theorem 6.2.9. As in the proof of Theorem 6.2.4, we need to show convergence of finite-dimensional distributions and tightness. We first prove the convergence of finite- dimensional distributions, that is, ZT(t)

f.d.d.

−→ Z(t) as T → ∞, where Z_T(t) and Z(t) are defined by (6.13) and (6.14), respectively.

By Lemma 6.3.10, the process ZT(t) defined in (6.13) has the same finite-dimensional

distributions as the process Z_T0 (t) defined in (6.33). Therefore, taking into account the linearity of multiple Wiener-Itˆo integral, and applying Cr´amer-Wold device, to prove

ZT(t) f.d.d.

−→ Z(t), it is enough to show that as T → ∞,

Ht,T(x1, x2) → Ht(x1, x2) in L2(R2), (6.41)

where Ht(x1, x2) and Ht,T(x1, x2) are as in (6.15) and (6.34), respectively.

First, we show pointwise convergence for a.e. (x1, x2) ∈ R2, that is,

Ht,T(x1, x2) = A1,T(x1, x2)|x1x2|−α/2

Z

R

∆t(x1+ u)∆t(x2− u)|u|−βA2,T(u)du (6.42)

→ H_t(x1, x2) = |x1x2|−α/2

Z

R

∆t(x1+ u)∆t(x2− u)|u|−βdu as T → ∞.

(6.43)

Because L1(x) is a slowly varying function, we have A1,T(x1, x2) → 1 as T → ∞, where

A1,T is as in (6.29). To show that the integral in (6.42) converges to the integral in

(6.43), note first that by (6.29), A2,T(u) → 1 as T → ∞ because L2(x) is a slowly varying

function. Hence one only needs to bound the integrand properly and apply the Dominated Convergence Theorem. To this end, observe that by (6.31) for T large enough, we have

gT(u; x1, x2) : = |∆t(x1+ u)||∆t(x2− u)||u|−βA2,T(u) (6.44)

≤ C|∆t(x1+ u)||∆t(x2− u)||u|−β(|u|+ |u|−) := g(u; x1, x2). (6.45)

By choosing  small enough, using Fubini Theorem and Lemma 6.3.8, we conclude that g(· ; x1, x2) ∈ L1(R) for a.e. (x1, x2) ∈ R2. Now (6.41) follows from (6.32) and the

Dominated Convergence Theorem.

To prove tightness, first observe that by the hypercontractivity inequality of the multiple Wiener-Itˆo integrals (see Major [2014], Corollary 5.6) and Lemma 6.3.11, for T large enough and for any 0 ≤ s ≤ t ≤ 1, there exists a constant C > 0 to satisfy

E|ZT(t) − ZT(s)|4 ≤ C2 E|ZT(t) − ZT(s)|2


2

where δ is a fixed number within the range 0 < 4δ < 2(α + β). Since by assumption α + β > 1/2, we can choose δ to satisfy 4δ > 1. Inequalities similar to (6.46) hold also for the limit process Z(t).

In view of (6.46) and a similar inequality for Z(t), it follows from Kolmogorov’s criterion (see, e.g., Bass [2011] Theorem 8.1(1)) that the processes ZT(t) and Z(t) admit continuous

versions when T is large enough.

Now the tightness of the family of measures generated by the processes {ZT(t) : T > 0}

Limit theorems for quadratic forms of L´evy-driven

continuous-time linear processes

We study the asymptotic behavior of a suitable normalized stochastic process {QT(t), t ∈

[0, 1]}. This stochastic process is generated by a Toeplitz type quadratic functional of a L´evy-driven continuous-time linear process. We show that under some Lp_{-type conditions}

imposed on the covariance function of the model and the kernel of the quadratic functional, the process QT(t) obeys a central limit theorem, that is, the finite-dimensional distributions

of the standard√T normalized process QT(t) tend to those of a normalized standard Brow-

nian motion. In contrast, when the covariance function of the model and the kernel of the quadratic functional have a slow power decay, then we have a non-central limit theorem for QT(t), that is, the finite-dimensional distributions of the process QT(t), normalized by Tγ

for some γ > 1/2, tend to those of a non-Gaussian non-stationary-increment self-similar process which can be represented by a double stochastic Wiener-Itˆo integral on R2_.

7.1 Introduction

Let {X(t), t ∈ R} be a L´evy-driven continuous-time stationary linear process defined by

X(t) = Z

R

where a(·) is a function from L2(R), and ξ(t) is a L´evy process satisfying the conditions: Eξ(t) = 0, Eξ2(1) = 1 and Eξ4(1) < ∞.

A L´evy process, {ξ(t), t ∈ R} is a process with independent and stationary increments, continuous in probability, with sample-paths which are right-continuous with left limits (c`adl`ag) and ξ(0) = ξ(0−) = 0. The Wiener process {B(t), t ≥ 0} and the centered Poisson process {N (t) − EN (t), t ≥ 0} are typical examples of centered L´evy processes. Notice that the covariance function of X(t) is given by

r(t) = EX(t)X(0) = Z

R

a(t + x)a(x)dx, (7.2)

and it possesses the spectral density

f (λ) = σ 2 2π|ba(λ)| 2₌ σ2 2π     Z R e−iλta(t)dt     2 , λ ∈ R. (7.3)

The function a(·) plays the role of a time-invariant filter.

Processes of the form (7.1) appear in many fields of science (economics, finance, physics, etc.), and cover a large class of popular models in continuous-time time series modeling. For instance, the so-called continuous-time autoregressive moving average (CARMA) mod- els, which are the continuous-time analogs of the classical autoregressive moving average (ARMA) models in discrete-time case, are of the form (7.1) and play a central role in the representation of continuous-time stationary time series. L´evy-driven CARMA processes permit the modelling of heavy-tailed and asymmetric time series and incorporate both distributional and sample-path information (see, e.g., Brockwell [2001, 2014]).

Consider the following Toeplitz type quadratic functional of the process X(u):

QT := Z T 0 Z T 0 b(u − v)X(u)X(v) du dv, T > 0, (7.4)

where

b(t) :=bg(t) = Z

R

eiλtg(λ)dλ, t ∈ R,

is the Fourier transform of some integrable even function g(λ), λ ∈ R. We will refer to g(λ) and to its Fourier transform b(t) as a generating function and generating kernel for the functional QT, respectively.

In this chapter we are interested in the asymptotic behavior as (T → ∞) of the stochas- tic process {QT(t), t ∈ [0, 1]}, generated by the functional QT:

QT(t) := Z T t 0 Z T t 0 b(u − v)X(u)X(v)dudv, t ∈ [0, 1]. (7.5)

Our goal is to establish functional limit theorems of the form 1

A(T )(QT(t) − EQT(t))

f.d.d.

−→ L(t), (7.6)

where A(T ) is a normalization factor, L(t) is the limit process, and the symbolf.d.d.−→ stands for convergence of finite-dimensional distributions.

Functionals of the form (7.5) and their discrete counterparts arise naturally in the statistical estimation of the spectrum of stationary processes. Limits such as (7.6) are necessary to establish asymptotic properties of these estimators (see, for example, Fox and Taqqu [1986], Ginovyan [2011], Giraitis et al. [2012], and references therein).

In the case where the underlying model {X(u), u ∈ R} is a Wiener-driven process, that is, X(u) is a Gaussian process, limit theorems of the form (7.4) were established in Bai et al. [2015], among others, where it was shown that if both the spectral density f of X(u) and the generating function g are regularly varying at the origin of orders α and β, respectively, then it is the sum α + β that determines the limiting process L(t). In fact, when

the limit process L(t) is a normalized standard Brownian motion, while when

α + β > 1/2,

the limit L(t) is a non-Gaussian self-similar process, which can be represented as a double Wiener-Itˆo integral on R2.

In this chapter, we consider the general case where the model {X(u), u ∈ R} is a continuous-time linear process driven from L´evy noise ξ(u) with time invariant filter a(·). Specifically, we show that under some Lp-type conditions imposed on the filter a(·) and the kernel b(·) of the quadratic functional, the process QT(t) obeys a central limit theorem, that

is, the finite-dimensional distributions of the standard√T normalized process QT(t) tend

to those of a normalized standard Brownian motion. In contrast, when the functions a(·) and b(·) have slow power decay, then we have a non-central limit theorem for QT(t), that

is, the finite-dimensional distributions of the process QT(t), normalized by Tγ for some

γ > 1/2, tend to those of a non-Gaussian non-stationary-increment self-similar process which can be represented by a double stochastic Wiener-Itˆo integral on R2.

We point out that our proofs of the central limit theorems are based on a new ap- proximation approach which reduces the quadratic integral form to a single integral form. This method can also be adapted to the discrete-time case. To prove the non-central limit theorems, we use the spectral representation of the underlying process, the properties of Wiener-Itˆo integrals, and a continuous analog of a method to establish convergence in dis- tribution of quadratic functionals to double Wiener-Itˆo integrals, developed by Surgailis [1982] (see also Giraitis et al. [2012]).

Limit theorems for quadratic forms of the type (7.5) have been considered by a number of authors, mostly for discrete-time stationary processes (see, e.g., Grenander and Szeg¨o [1958], Fox and Taqqu [1985, 1987], Giraitis and Surgailis [1990], Terrin and Taqqu [1990], Giraitis and Taqqu [1999], Ginovyan and Sahakyan [2005], and references therein). The continuous-time case where X(t) is Gaussian has been mainly considered in Ginovian

[1994], Ginovyan and Sahakyan [2007], and Bai et al. [2015].

To the best of our knowledge, the only work addressing the quadratic functionals of the L´evy-driven continuous-time linear process X(t) is Avram et al. [2010], where a central limit theorem for the quadratic functional (7.4) was stated (without proof) under some Lp- type conditions imposed on the spectral density f (λ) of X(u) and the generating function g(λ) (see Remark 7.2.6 below). For a related study of the sample covariances of L´evy-driven moving average processes we refer to the recent papers by Cohen and Lindner [2013], and Spangenberg [2015].

In our setting, where the underlying process X(t) is not necessarily Gaussian, additional complications arise due to the contribution of the random diagonal term in the double stochastic integral with respect to L´evy noise, which is not present in the case of Gaussian noise (see Remark 7.2.3 below).

The chapter is organized as follows. In Section 7.2 we state the main results of the chapter. In Section 7.3 we give a number of preliminary results that are used in the proofs of the main results. Sections 7.4 and 7.5 contain the proofs of the main results.