An adaptive estimator of the memory parameter and the goodness-of-fit test using a multidimensional increment ratio statistic

The increment ratio (IR) statistic was first defined and studied in Surgailis et al. (2007) [19] for estimating the memory parameter either of a stationary or an increment stationary Gaussian process. Here three extensions are proposed in the case of stationary processes. First, a multidimensional central limit theorem is established for a vector composed by several IR statistics. Second, a goodness-of-fit χ²-type test can be deduced from this theorem. Finally, this theorem allows to construct adaptive versions of the estimator and the test which are studied in a general semiparametric frame. The adaptive estimator of the long-memory parameter is proved to follow an oracle property. Simulations attest to the interesting accuracies and robustness of the estimator and the test, even in the non Gaussian case.     

Stationarity and Self-Similarity Characterization of the Set-Indexed Fractional Brownian Motion

The set-indexed fractional Brownian motion (sifBm) has been defined by Herbin–Merzbach (J. Theor. Probab. 19(2):337–364, 2006) for indices that are subsets of a metric measure space. In this paper, the sifBm is proved to satisfy a strengthened definition of increment stationarity. This new definition for stationarity property allows us to get a complete characterization of this process by its fractal properties: The sifBm is the only set-indexed Gaussian process which is self-similar and has stationary increments. Using the fact that the sifBm is the only set-indexed process whose projection on any increasing path is a one-dimensional fractional Brownian motion, the limitation of its definition for a self-similarity parameter 0<H<1/2 is studied, as illustrated by some examples. When the indexing collection is totally ordered, the sifBm can be defined for 0<H<1.     

Anticipative calculus for Lévy processes and stochastic differential equations*

We develop an anticipative calculus for Lévy processes with finite second moment for analysing anticipating stochastic differential equations. The calculus is based on the chaos expansion of square-integrable random variables in terms of iterated integrals with respect to the compensated Poisson random measure. We define a space of smooth and generalized random variables in terms of such chaos expansions, and present anticipative stochastic integration, the Wick product and the so-called 𝒮-transform. These concepts serve as tools for studying general Wick type stochastic differential equations with anticipative initial conditions. We apply the 𝒮-transform to find the unique solutions to a class of linear stochastic differential equations. The solutions can be expressed in terms of the Wick product.     

On asymptotic normality of estimates for correlation functions of stationary Gaussian processes in the space of continuous functions

We establish conditions of the weak convergence of the empirical correlogram of a stationary Gaussian process to some Gaussian process in the space of continuous functions. We prove that such a convergence holds for a broad class of stationary Gaussian processes with square integrable spectral density.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 47, No. 11, pp. 1485–1497, November, 1995.This work was financially supported by the Ukrainian State Committee on Science and Technology.     

Asymptotic Behavior of Gaussian Samples in Fréchet Spaces

Abstract

In this article we investigate unnormalized samples of Gaussian random elements in a separable Fréchet space 𝕄. First we describe a connection between shifts of a Gaussian measure μ in a separable Fréchet space and the infinite product of standard normal distributions in ?^∞, and on the basis of this result we derive the so‐called self‐sufficient expansion for Gaussian random elements in a Fréchet space. Moreover, we find lower bounds for the Gaussian measure μ of shifted balls in 𝕄 and estimate the metric entropy of balls in the Hilbert space ? ? 𝕄 which generates μ. Finally, applying the Brunn–Minkowski inequality we prove a kind of the logarithmic law of large numbers. The last result is an extension of the analogous theorem obtained by Goodman (Characteristics of normal samples. Ann. Probab. 1988, 16, 1281–1290), for a sequence of Gaussian random elements in a separable Banach space.     

Dependence of polynomial chaos on random types of forces of KdV equations

In this study, one-dimensional stochastic Korteweg–de Vries equation with uncertainty in its forcing term is considered. Extending the Wiener chaos expansion, a numerical algorithm based on orthonormal polynomials from the Askey scheme is derived. Then dependence of polynomial chaos on the distribution type of the random forcing term is inspected. It is numerically shown that when Hermite (Laguerre or Jacobi) polynomial chaos is chosen as a basis in the Gaussian (Gamma or Beta, respectively) random space for uncertainty, the solution to the KdV equation converges exponentially. If a proper polynomial chaos is not used, however, the solution converges with slower rate.     

Existence of the linear prediction for Banach space valued Gaussian processes

The correspondence between Gaussian stochastic processes with values in a Banach space E and cylindrical processes which are related to them is studied. It is shown that the linear prediction of an E-valued Gaussian process is an E-valued random variable as well as the spectral measure of an E-valued Gaussian stationary process is a Gaussian random measure.     

Operator differential-algebraic equations with noise arising in fluid dynamics

We study linear semi-explicit stochastic operator differential algebraic equations (DAEs) for which the constraint equation is given in an explicit form. In particular, this includes the Stokes equations arising in fluid dynamics. We combine a white noise polynomial chaos expansion approach to include stochastic perturbations with deterministic regularization techniques. With this, we are able to include Gaussian noise and stochastic convolution terms as perturbations in the differential as well as in the constraint equation. By the application of the polynomial chaos expansion method, we reduce the stochastic operator DAE to an infinite system of deterministic operator DAEs for the stochastic coefficients. Since the obtained system is very sensitive to perturbations in the constraint equation, we analyze a regularized version of the system. This then allows to prove the existence and uniqueness of the solution of the initial stochastic operator DAE in a certain weighted space of stochastic processes.     

The behaviour of a non-differentiable stationary Gaussian process after a level crossing

A model process is obtained for the behaviour of a non-differentiable but continuous stationary Gaussian process after a level crossing. It is shown that the sampled process conditioned on a crossing of a fixed level in Slepian's sense, converges weakly towards the model process, when the sample distance decreases to zero. Further it is noticed that there is no difference between conditioning on a vertical window and on a horizontal window in this case.     

Information in a scheme with additive noise

A formula is proved expressing the information contained in a stationary, linearly regular, Gaussian process with an independent additive increment relative to the original unperturbed process.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova Akad. Nauk SSSR, Vol. 55, pp. 117–127, 1976.     

Simulation accuracy of stationary Gaussian stochastic processes inL 2(0,T)

Models of stationary Gaussian stochastic processes with discrete and continuous spectra are constructed. Simulation of stationary Gaussian processes with a continuous spectrum is considered for the following cases: when the covariance function of the stochastic process is expandable in a Fourier series with positive coefficients; when the spectrum of the stationary Gaussian stochastic process is concentrated on the interval [0, ]; and in the general case. The stationary Gaussian process is simulated with prescribed reliability and accuracy in L²(0, T).Kiev University. Translated from Vychislitel'naya i Prikladnaya Matematika, No. 75, pp. 108–115, 1991.     

Stochastic Response of a Continuous System with Stochastic Surface Irregularities to an Accelerated Load

The problem of calculating the second moment characteristics of the response of a general class of nonconservative linear distributed parameter systems with stochastically varying surface roughness, excited by a moving concentrated load, is investigated. In particular the case of an accelerated load is discussed. The surface roughness is modeled as a Gaussian stationary second order process. For the stochastic representation of the surface roughness a orthogonal series expansion of the covariance kernel, the so called Karhunen‐Loéve expansion, is applied. The resulting initial/boundary value problem is transformed by eigenfunction expansion into the modal state space. Second moment characteristics of the response are determined numerically by direct integration using a Runge‐Kutta method.     

Cycle Range Distributions for Gaussian Processes Exact and Approximative Results

Wave cycles, i.e. pairs of local maxima and minima, play an important role in many engineering fields. Many cycle definitions are used for specific purposes, such as crest–trough cycles in wave studies in ocean engineering and rainflow cycles for fatigue life predicition in mechanical engineering. The simplest cycle, that of a pair of local maximum and the following local minimum is also of interest as a basis for the study of more complicated cycles. This paper presents and illustrates modern computational tools for the analysis of different cycle distributions for stationary Gaussian processes with general spectrum. It is shown that numerically exact but slow methods will produce distributions in almost complete agreement with simulated data, but also that approximate and quick methods work well in most cases. Of special interest is the dependence relation between the cycle average and the cycle range for the simple maximum–minimum cycle and its implication for the range distribution. It is observed that for a Gaussian process with rectangular box spectrum, these quantities are almost independent and that the range is not far from a Rayleigh distribution. It will also be shown that had there been a Gaussian process where exact independence hold then the range would have had an exact Rayleigh distribution. Unfortunately no such Gaussian process exists.This revised version was published online in March 2005 with corrections to the cover date.     

On large deviations in testing simple hypotheses for locally stationary Gaussian processes

We derive a large deviation result for the log-likelihood ratio for testing simple hypotheses in locally stationary Gaussian processes. This result allows us to find explicitly the rates of exponential decay of the error probabilities of type I and type II for Neyman?CPearson tests. Furthermore, we obtain the analogue of classical results on asymptotic efficiency of tests such as Stein??s lemma and the Chernoff bound, as well as the more general Hoeffding bound concerning best possible joint exponential rates for the two error probabilities.     

On the goodness-of-fit testing for a switching diffusion process

We study the asymptotic behavior of the Cramér–von Mises type statistic in the goodness-of-fit hypotheses testing problem for ergodic diffusion processes. The basic (simple) hypothesis is defined by the stochastic differential equation with sign-type trend coefficient and known diffusion coefficient. It is shown that the limit distribution of the proposed test statistic (under hypothesis) is defined by the integral type functional of continuous Gaussian process. We provide the Karhunen–Loève expansion of the corresponding limiting process and show that the eigenfunctions in this expansion are expressed in terms of Bessel functions. This representation for the limit statistic allows us to approximate the threshold.     

Are the eigenvalues of preconditioned banded symmetric Toeplitz matrices known in almost closed form?

The problem of computing oscillatory integrals with general oscillators is considered. We employ a Filon-type method, where the interpolation basis functions are chosen in such a way that the moments are in terms of elementary functions and the oscillator only. This allows us to evaluate the moments rapidly and easily without needing to engage hypergeometric functions. The proposed basis functions form a Chebyshev set for any oscillator function even if it has some stationary points in the integration interval. This property enables us to employ the Filon-type method without needing any information about the stationary points if any. Interpolation by the proposed basis functions at the Fekete points (which are known as nearly optimal interpolation points), when combined with the idea of splines, leads to a reliable convergent method for computing the oscillatory integrals. Our numerical experiments show that the proposed method is more efficient than the earlier ones with the same advantages.     

Asymptotic properties of correlation bounds in functional spaces. II

This paper is an extension of [11]. Starting from the results of our first paper we prove by inclusion theorems that bounds for the correlation function of a stationary Gaussian process in the space of continuous functions with weight are strongly consistent and asymptotically normal. We construct the simplest functional confidence intervals in these spaces for the indeterminate correlation function.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 43, No. 3, pp. 322–329, March, 1991.     

Asymptotic properties of correlation bounds in functional spaces. II

This paper is an extension of [11]. Starting from the results of our first paper we prove by inclusion theorems that bounds for the correlation function of a stationary Gaussian process in the space of continuous functions with weight are strongly consistent and asymptotically normal. We construct the simplest functional confidence intervals in these spaces for the indeterminate correlation function.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 43, No. 3, pp. 322–329, March, 1991.     

An estimate of the rate of convergence in the Poisson approximation theorem for large excursions of a Gaussian stationary process

We prove a theorem on the rate of convergence to a Poisson process of the pointwise process of the number of high-level crossings by a twice-differentiable Gaussian stationary process. We study the pathwise convergence of pointwise processes of crossing type by the single probability space method.Translated fromTeoriya Sluchainykh Protsessov, Vol. 15, pp. 74–79, 1987.     

Turbulent chaos

A new definition of a chaotic invariant set for a continuous semi-flow in a metric space is given. This definition generalizes the known definition of Devaney and allows us to take into account a specific feature arising in a noncompact and infinite-dimensional case, the so-called turbulent chaos.