Nonparametric estimation of the local Hurst function of multifractional Gaussian processes

A new nonparametric estimator of the local Hurst function of a multifractional Gaussian process based on the increment ratio (IR) statistic is defined. In a general frame, the point-wise and uniform weak and strong consistency and a multidimensional central limit theorem for this estimator are established. Similar results are obtained for a refinement of the generalized quadratic variations (QV) estimator. The example of the multifractional Brownian motion is studied in detail. A simulation study is included showing that the IR-estimator is more accurate than the QV-estimator.     

Central Limit Theorems for approximate quadratic variations of pure jump Itô semimartingales

We derive Central Limit Theorems for the convergence of approximate quadratic variations, computed on the basis of regularly spaced observation times of the underlying process, toward the true quadratic variation. This problem was solved in the case of an Itô semimartingale having a non-vanishing continuous martingale part. Here we focus on the case where the continuous martingale part vanishes and find faster rates of convergence, as well as very different limiting processes.     

A wavelet analysis of the Rosenblatt process: Chaos expansion and estimation of the self-similarity parameter

By using chaos expansion into multiple stochastic integrals, we make a wavelet analysis of two self-similar stochastic processes: the fractional Brownian motion and the Rosenblatt process. We study the asymptotic behavior of the statistic based on the wavelet coefficients of these processes. Basically, when applied to a non-Gaussian process (such as the Rosenblatt process) this statistic satisfies a non-central limit theorem even when we increase the number of vanishing moments of the wavelet function. We apply our limit theorems to construct estimators for the self-similarity index and we illustrate our results by simulations.     

On quadratic functionals of the Brownian sheet and related processes

Motivated by asymptotic problems in the theory of empirical processes, and specifically by tests of independence, we study the law of quadratic functionals of the (weighted) Brownian sheet and of the bivariate Brownian bridge on [0,1]²$[0,1{]}^2$. In particular: (i) we use Fubini-type techniques to establish identities in law with quadratic functionals of other Gaussian processes, (ii) we explicitly calculate the Laplace transform of such functionals by means of Karhunen–Loève expansions, (iii) we prove central and non-central limit theorems in the spirit of Peccati and Yor [Four limit theorems involving quadratic functionals of Brownian motion and Brownian bridge, Asymptotic Methods in Stochastics, American Mathematical Society, Fields Institute Communication Series, 2004, pp. 75–87] and Nualart and Peccati [Central limit theorems for sequences of multiple stochastic integrals, Ann. Probab. 33(1) (2005) 177–193]. Our results extend some classical computations due to Lévy [Wiener's random function and other Laplacian random functions, in: Second Berkeley Symposium in Probability and Statistics, 1950, pp. 171–186], as well as the formulae recently obtained by Deheuvels and Martynov [Karhunen–Loève expansions for weighted Wiener processes and Brownian bridges via Bessel functions, Progress in Probability, vol. 55, Birkhäuser Verlag, Basel, 2003, pp. 57–93].     

Central limit theorems for multiple stochastic integrals and Malliavin calculus

We give a new characterization for the convergence in distribution to a standard normal law of a sequence of multiple stochastic integrals of a fixed order with variance one, in terms of the Malliavin derivatives of the sequence. We also give a new proof of the main theorem in [D. Nualart, G. Peccati, Central limit theorems for sequences of multiple stochastic integrals, Ann. Probab. 33 (2005) 177–193] using techniques of Malliavin calculus. Finally, we extend our result to the multidimensional case and prove a weak convergence result for a sequence of square integrable random vectors, giving an application.     

Almost sure invariance principles via martingale approximation

In this paper, we estimate the rest of the approximation of a stationary process by a martingale in terms of the projections of partial sums. Then, based on this estimate, we obtain almost sure approximation of partial sums by a martingale with stationary differences. The results are exploited to further investigate the central limit theorem and its invariance principle started at a point, the almost sure central limit theorem, as well as the law of the iterated logarithm via almost sure approximation with a Brownian motion, improving the results available in the literature. The conditions are well suited for a variety of examples; they are easy to verify, for instance, for linear processes and functions of Bernoulli shifts.     

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$q\ge 1$ and q+1$q+1$ 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.     

An infinite dimensional convolution theorem with applications to the efficient estimation of the integrated volatility

This paper proposes a general approach to obtain asymptotic lower bounds for the estimation of random functionals. The main result is an abstract convolution theorem in a non parametric setting, based on an associated LAMN property. This result is then applied to the estimation of the integrated volatility, or related quantities, of a diffusion process, when the diffusion coefficient depends on an independent Brownian motion.     

Path regularity and explicit convergence rate for BSDE with truncated quadratic growth

We consider backward stochastic differential equations with drivers of quadratic growth (qgBSDE). We prove several statements concerning path regularity and stochastic smoothness of the solution processes of the qgBSDE, in particular we prove an extension of Zhang’s path regularity theorem to the quadratic growth setting. We give explicit convergence rates for the difference between the solution of a qgBSDE and its truncation, filling an important gap in numerics for qgBSDE. We give an alternative proof of second order Malliavin differentiability for BSDE with drivers that are Lipschitz continuous (and differentiable), and then derive an analogous result for qgBSDE.     

Fourier transform methods for pathwise covariance estimation in the presence of jumps

We provide a new non-parametric Fourier procedure to estimate the trajectory of the instantaneous covariance process (from discrete observations of a multidimensional price process) in the presence of jumps extending the seminal work of Malliavin and Mancino (2002, 2009). Our approach relies on a modification of (classical) jump-robust estimators of integrated realized covariance to estimate the Fourier coefficients of the covariance trajectory. Using Fourier–Féjer inversion we reconstruct the path of the instantaneous covariance. We prove consistency and a central limit theorem (CLT) and in particular that the asymptotic estimator variance is smaller by a factor 2/3 in comparison to classical local estimators.     

On the convergence to the multiple Wiener-Itô integral

We study the convergence to the multiple Wiener-Itô integral from processes with absolutely continuous paths. More precisely, consider a family of processes, with paths in the Cameron-Martin space, that converges weakly to a standard Brownian motion in C₀([0,T]). Using these processes, we construct a family that converges weakly, in the sense of the finite dimensional distributions, to the multiple Wiener-Itô integral process of a function f∈L²(n[0,T]). We prove also the weak convergence in the space C₀([0,T]) to the second-order integral for two important families of processes that converge to a standard Brownian motion.     

Some remarks on a question of Strassen

Summary Strassen's original functional law of the iterated logarithm for partial sums and Brownian motion examined convergence and clustering in the sup-norm. Here we address what happens if we use the much larger H-norm. We provide the answer to a query which appeared at the end of Strassen's original paper, and also present several contrasting results which are shown to be essentially best possible.Supported in part by NSA Grant MDA-904-93-H-3033Supported in part by NSF Grant DMS-9400024     

Embedding and asymptotic expansions for martingales

The paper develops a way of embedding general martingales in continuous ones in such a way that the quadratic variation of the continuous martingale has conditional cumulants (given the original martingale) that are explicitly given in terms of optional and predictable variations of the original process. Bartlett identities for the conditional cumulants are also found. A main corollary to these results is the establishment of second (and in some cases higher) order asymptotic expansions for martingales.Research supported in part by National Science Foundation grant DMS 93-05601 and Army Research Office grant DAAH04-1-0105     

Sudakov's typical marginals, random linear functionals and a conditional central limit theorem

Summary. V.N. Sudakov [Sud78] proved that the one-dimensional marginals of a high-dimensional second order measure are close to each other in most directions. Extending this and a related result in the context of projection pursuit of P. Diaconis and D. Freedman [Dia84], we give for a probability measure and a random (a.s.) linear functional on a Hilbert space simple sufficient conditions under which most of the one-dimensional images of under are close to their canonical mixture which turns out to be almost a mixed normal distribution. Using the concept of approximate conditioning we deduce a conditional central limit theorem (theorem 3) for random averages of triangular arrays of random variables which satisfy only fairly weak asymptotic orthogonality conditions. Received: 25 July 1995 / In revised form: 20 June 1996     

On the deviation between the distributions of sums and maximum sums of IID random vectors

Summary For a sequence of independent and identically distributed random vectors, with finite moment of order less than or equal to the second, the rate at which the deviation between the distribution functions of the vectors of partial sums and maximums of partial sums is obtained both when the sample size is fixed and when it is random, satisfying certain regularity conditions. When the second moments exist the rate is of ordern ^−1/4 (in the fixed sample size case). Two applications are given, first, we compliment some recent work of Ahmad (1979,J. Multivariate Anal.,9, 214–222) on rates of convergence for the vector of maximum sums and second, we obtain rates of convergence of the concentration functions of maximum sums for both the fixed and random sample size cases.     

Winding numbers for 2-dimensional,positive recurrent diffusions

A well-known theorem by Spitzer states that the winding number of a standard Brownian motion around the origin is asymptotically Cauchy-distributed. A similar result is derived for positive recurrent diffusions in the plane given by a non-degenerate stochastic equation.     

Local limit theorems for multiplicative free convolutions

This paper describes the quality of convergence to an infinitely divisible law relative to free multiplicative convolution. We show that convergence in distribution for products of identically distributed and infinitesimal free random variables implies superconvergence of their probability densities to the density of the limit law. Superconvergence to the marginal law of free multiplicative Brownian motion at a specified time is also studied. In the unitary case, the superconvergence to free Brownian motion and that to the Haar measure are shown to be uniform over the entire unit circle, implying further a free entropic limit theorem and a universality result for unitary free Lévy processes. Finally, the method of proofs on the positive half-line gives rise to a new multiplicative Boolean to free Bercovici–Pata bijection.     

Change point estimators by local polynomial fits under a dependence assumption

We study a random design regression model generated by dependent observations, when the regression function itself (or its ν-th derivative) may have a change or discontinuity point. A method based on the local polynomial fits with one-sided kernels to estimate the location and the jump size of the change point is applied in this paper. When the jump location is known, a central limit theorem for the estimator of the jump size is established; when the jump location is unknown, we first obtain a functional limit theorem for a local dilated-rescaled version estimator of the jump size and then give the asymptotic distributions for the estimators of the location and the jump size of the change point. The asymptotic results obtained in this paper can be viewed as extensions of corresponding results for independent observations. Furthermore, a simulated example is given to show that our theory and method perform well in practice.