Hermite Process Research Articles

Tempered Hermite process

A tempered Hermite process modifies the power law kernel in the time domain representation of a Hermite process by multiplying an exponential tempering factor $\lambda>0$ such that the process is well defined for Hurst parameter $H>\frac{1}{2}$. A tempered Hermite process is the weak convergence limit of a certain discrete chaos process.

Generalized continuous time random walks and Hermite processes

Generalized continuous time random walks with independent, heavy-tailed random waiting times and long range dependent jumps are considered. Their scaling limits are determined in terms of the Hermite processes and inverse of stable subordinators. These limiting processes provide an interesting new class of non-Markovian, non-Gaussian self-similar processes.Tail probability estimates for the limiting process are established, which are applied in turn to establish uniform and local moduli of continuity.

Weak convergence of the weighted sequential empirical process of some long-range dependent data

Let (Xk)k≥1 be a Gaussian long-range dependent process with EX1=0, EX12=1 and covariance function r(k)=k−DL(k). For any measurable function G let (Yk)k≥1=(G(Xk))k≥1. We study the asymptotic behaviour of the associated sequential empirical process (RN(x,t)) with respect to a weighted sup-norm ‖⋅‖w. We show that, after an appropriate normalization, (RN(x,t)) converges weakly in the space of cádlág functions with finite weighted norm to a Hermite process.

Structure of the third moment of the generalized Rosenblatt distribution

The Rosenblatt distribution appears as limit in non-central limit theorems. The generalized Rosenblatt distribution is obtained by allowing different power exponents in the kernel that defines the usual Rosenblatt distribution. We derive an explicit formula for its third moment, correcting the one in Maejima and Tudor (2012) and Tudor (2013). Evaluating this formula numerically, we are able to confirm that the class of generalized Hermite processes is strictly richer than the class of Hermite processes.

Asymptotic behavior of the quadratic variation of the sum of two Hermite processes of consecutive orders

Hermite processes are self-similar processes with stationary increments which appear as limits of normalized sums of random variables with long range dependence. The Hermite process of order 1 is fractional Brownian motion and the Hermite process of order 2 is the Rosenblatt process. We consider here the sum of two Hermite processes of orders q≥1 and q+1 and of different Hurst parameters. We then study its quadratic variations at different scales. This is akin to a wavelet decomposition. We study both the cases where the Hermite processes are dependent and where they are independent. In the dependent case, we show that the quadratic variation, suitably normalized, converges either to a normal or to a Rosenblatt distribution, whatever the order of the original Hermite processes.

On a class of self-similar processes with stationary increments in higher order Wiener chaoses

We study a class of self-similar processes with stationary increments belonging to higher order Wiener chaoses which are similar to Hermite processes. We obtain an almost sure wavelet-like expansion of these processes. This allows us to compute the pointwise and local Hölder regularity of sample paths and to analyse their behaviour at infinity. We also provide some results on the Hausdorff dimension of the range and graphs of multidimensional anisotropic self-similar processes with stationary increments defined by multiple Wiener–Itô integrals.

Generalized Hermite processes, discrete chaos and limit theorems

We introduce a broad class of self-similar processes {Z(t),t≥0} called generalized Hermite processes. They have stationary increments, are defined on a Wiener chaos with Hurst index H∈(1/2,1), and include Hermite processes as a special case. They are defined through a homogeneous kernel g, called the “generalized Hermite kernel”, which replaces the product of power functions in the definition of Hermite processes. The generalized Hermite kernels g can also be used to generate long-range dependent stationary sequences forming a discrete chaos process {X(n)}. In addition, we consider a fractionally-filtered version Zβ(t) of Z(t), which allows H∈(0,1/2). Corresponding non-central limit theorems are established. We also give a multivariate limit theorem which mixes central and non-central limit theorems.

MULTIVARIATE LIMIT THEOREMS IN THE CONTEXT OF LONG‐RANGE DEPENDENCE

We study the limit law of a vector made up of normalized sums of functions of long‐range dependent stationary Gaussian series. Depending on the memory parameter of the Gaussian series and on the Hermite ranks of the functions, the resulting limit law may be (a) a multi‐variate Gaussian process involving dependent Brownian motion marginals, (b) a multi‐variate process involving dependent Hermite processes as marginals or (c) a combination. We treat cases (a) and (b) in general and case (c) when the Hermite components involve ranks 1 and 2. We include a conjecture about case (c) when the Hermite ranks are arbitrary, although the conjecture can be resolved in some special cases.

Non-central limit theorem of the weighted power variations of Gaussian processes

By using the techniques of the Malliavin calculus, we investigate the asymptotic behavior of the weighted q-variations of continuous Gaussian process of the form Bt=∫0tK(t,s)dW(s), where W is the standard Brownian motion and K is a square integrable kernel. In particular, in the case of fractional Brownian motion with the Hurst parameter H, the limit can be expressed as the sum of q+1 Skorohod integrals of the Hermite process with self-similarity q(H−1)+1. This result gives the relation between the Skorohod integral and a pathwise Young integral of the Hermite process.

On piecewise polynomial regression under general dependence conditions, with an application to calcium-imaging data

Motivated by the analysis of glomerular time series extracted from calcium-imaging data, asymptotic theory for piecewise polynomial and spline regression with partially free knots and residuals exhibiting three types of dependence structures (long memory, short memory and anti-persistence) is considered. Unified formulas based on fractional calculus are derived for subordinated residual processes in the domain of attraction of a Hermite process. The results are applied to testing for the effect of a neurotransmitter on the response of olfactory neurons in honeybees to odorant stimuli.

Central and Non-central Limit Theorems in a Free Probability Setting

Long-range dependence in time series may yield non-central limit theorems. We show that there are analogous time series in free probability with limits represented by multiple Wigner integrals, where Hermite processes are replaced by non-commutative Tchebycheff processes. This includes the non-commutative fractional Brownian motion and the non-commutative Rosenblatt process.

HERMITE VARIATIONS OF THE FRACTIONAL BROWNIAN SHEET

We prove central and non-central limit theorems for the Hermite variations of the anisotropic fractional Brownian sheet Wα, βwith Hurst parameter (α, β) ∈ (0, 1)2. When [Formula: see text] or [Formula: see text] a central limit theorem holds for the renormalized Hermite variations of order q ≥ 2, while for [Formula: see text] we prove that these variations satisfy a non-central limit theorem. In fact, they converge to a random variable which is the value of a two-parameter Hermite process at time (1, 1).

Dissipative Stochastic Evolution Equations Driven by General Gaussian and Non-Gaussian Noise

We study a class of stochastic evolution equations with a dissipative forcing nonlinearity and additive noise. The noise is assumed to satisfy rather general assumptions about the form of the covariance function; our framework covers examples of Gaussian processes, like fractional and bifractional Brownian motion and also non Gaussian examples like the Hermite process. We give an application of our results to the study of the stochastic version of a common model of potential spread in a dendritic tree. Our investigation is specially motivated by possibility to introduce long-range dependence in time of the stochastic perturbation.

Self-similarity parameter estimation and reproduction property for non-Gaussian Hermite processes

Let (Z (q;H) t )t2(0;1) be a Hermite processes of order q and with Hurst parameter H 2 ( 1 ;1). This process is H-self-similar, it has stationary increments and it exhibits long-range dependence. This class contains the fractional Brownian motion (for q = 1) and the Rosenblatt process (for q = 2). We study in this paper the variations of Z (q;H) by using multiple Wiener -It^o stochastic integrals and Malliavin calculus. We prove a reproduction property for this class of processes in the sense that the terms appearing in the chaotic decomposition of the their variations give birth to other Hermite processes of dierent orders and with dierent Hurst parameters. We apply our results to construct a consistent estimator for the self-similarity parameter from discrete observations of a Hermite process.

Regularization and integral representations of Hermite processes

It is known that Hermite processes have a finite-time interval representation. For fractional Brownian motion, the representation has been well known and plays a fundamental role in developing stochastic calculus for the process. For the Rosenblatt process, the finite-time interval representation was originally established by using cumulants. The representation was extended to general Hermite processes through the convergence of suitable partial sum processes. We provide here an alternative and different proof for the finite-time interval representation of Hermite processes. The approach is based on regularization of Hermite processes and the fractional Gaussian noises underlying them, and does not use cumulants nor convergence of partial sums.

Central and non-central limit theorems for weighted power variations of fractional Brownian motion

Dans ce papier, nous prouvons des théorèmes de la limite centrale et non-centrale pour les variations à poids d’ordre q du mouvement brownien fractionnaire d’indice H∈(0, 1), pour q un entier supérieur ou égal à 2. Il y a trois cas, suivant la position de H par rapport à 1/2q et 1−1/2q. Si 1/2q<H≤1−1/2q, nous montrons un théorème de la limite centrale vers une variable aléatoire de loi conditionnellement gaussienne. Si H<1/2q, nous montrons la convergence dans L2 vers une limite qui dépend seulement du mouvement brownien fractionnaire. Si H>1−1/2q, nous montrons la convergence dans L2 vers une intégrale stochastique par rapport au processus d’Hermite d’ordre q.

Application of Malliavin calculus to long-memory parameter estimation for non-Gaussian processes

Using multiple Wiener–Itô stochastic integrals and Malliavin calculus we study the rescaled quadratic variations of a general Hermite process of order q with long-memory (Hurst) parameter H ∈ ( 1 2 , 1 ) . We apply our results to the construction of a strongly consistent estimator for H. It is shown that the estimator is asymptotically non-normal, and converges in the mean-square, after normalization, to a standard Rosenblatt random variable. To cite this article: A. Chronopoulou et al., C. R. Acad. Sci. Paris, Ser. I 347 (2009).

Integrated functionals of normal and fractional processes

Consider Ztf(u)=∫0tuf(Ns) ds, t>0, u∈[0, 1], where N=(Nt)t∈ℝ is a normal process and f is a measurable real-valued function satisfying Ef(N0)2<∞ and Ef(N0)=0. If the dependence is sufficiently weak Hariz [J. Multivariate Anal. 80 (2002) 191–216] showed that Ztf/t1/2 converges in distribution to a multiple of standard Brownian motion as t→∞. If the dependence is sufficiently strong, then Zt/(EZt(1)2)1/2 converges in distribution to a higher order Hermite process as t→∞ by a result by Taqqu [Wahrsch. Verw. Gebiete 50 (1979) 53–83]. When passing from weak to strong dependence, a unique situation encompassed by neither results is encountered. In this paper, we investigate this situation in detail and show that the limiting process is still a Brownian motion, but a nonstandard norming is required. We apply our result to some functionals of fractional Brownian motion which arise in time series. For all Hurst indices H∈(0, 1), we give their limiting distributions. In this context, we show that the known results are only applicable to H<3/4 and H>3/4, respectively, whereas our result covers H=3/4.

Nonstandard limit theorem for infinite variance functionals

We consider functionals of long-range dependent Gaussian sequences with infinite variance and obtain nonstandard limit theorems. When the long-range dependence is strong enough, the limit is a Hermite process, while for weaker long-range dependence, the limit is α-stable Lévy motion. For the critical value of the long-range dependence parameter, the limit is a sum of a Hermite process and α-stable Lévy motion.

Wiener Integrals with Respect to the Hermite Process and a Non-Central Limit Theorem

We introduce Wiener integrals with respect to the Hermite process and we prove a non-central limit theorem in which this integral appears as limit. As an example, we study a generalization of the fractional Ornstein–Uhlenbeck process.