Exact L 2 Small Balls of Gaussian Processes

We prove a comparison theorem extending Li(6) and develop a complex-analytic approach to treat L ² small ball probabilities of Gaussian processes. We demonstrate the techniques for the m-times integrated Brownian motions and in examples where one can not apply Li comparison theorem.     

Logarithmic L 2-Small Ball Asymptotics with Respect to a Self-Similar Measure for Some Gaussian Processes

We find logarithmic small ball asymptotics for the L₂-norm with respect to self-similar measures for a certain class of Gaussian processes including Brownian motion, Ornstein-Uhlenbeck process, and their integrated counterparts. Bibliography: 46 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 311, 2004, pp. 190–213.     

Small Deviations for Gaussian Markov Processes Under the Sup-Norm

Let {X(t); 0t1} be a real-valued continuous Gaussian Markov process with mean zero and covariance (s, t) = EX(s) X(t) 0 for 0<s, t<1. It is known that we can write (s, t) = G(min(s, t)) H(max(s, t)) with G>0, H>0 and G/H nondecreasing on the interval (0, 1). We show that

Functional Quantization and Small Ball Probabilities for Gaussian Processes

Quantization consists in studying the L ^r -error induced by the approximation of a random vector X by a vector (quantized version) taking a finite number n of values. We investigate this problem for Gaussian random vectors in an infinite dimensional Banach space and in particular, for Gaussian processes. A precise link proved by Fehringer⁽⁴⁾ and Dereich et al. ⁽³⁾ relates lower and upper bounds for small ball probabilities with upper and lower bounds for the quantization error, respectively. We establish a complete relationship by showing that the same holds for the direction from the quantization error to small ball probabilities. This allows us to compute the exact rate of convergence to zero of the minimal L ^r -quantization error from logarithmic small ball asymptotics and vice versa.     

On the Sharp Constant in the Small Ball Asymptotics of Some Gaussian Processes under L 2-Norm

We find the sharp constant in the small L ₂-deviation asymptotics for a wide class of Gaussian processes including the m-times integrated Wiener process and the m-times integrated Ornstein–Uhlenbeck process. Extremal properties of usual and Euler integration are proved. Bibliography: 19 titles.     

On the Equivalence of Multiparameter Gaussian Processes

Our purpose is to characterize the multiparameter Gaussian processes, that is Gaussian sheets, that are equivalent in law to the Brownian sheet and to the fractional Brownian sheet. We survey multiparameter analogues of the Hitsuda, Girsanov and Shepp representations. As an application, we study a special type of stochastic equation with linear noise.      

Chung-type Law of the Iterated Logarithm on l^P-valued Gaussian Processes

By estimating small ball probabilities for l^P-valued Gaussian processes, a Chung-type law of the iterated logarithm of l^P-valued Gaussian processes is given.     

Exact Asymptotics of the Density of the Transition Probability for Discontinuous Markov Processes

In this paper we consider some Kolmogorov–Feller equations with a small parameter h. We present a method for constructing the exact (exponential) asymptotics of the fundamental solution of these equations for finite time intervals uniformly with respect to h. This means that we construct an asymptotics of the density of the transition probability for discontinuous Markov processes. We justify the asymptotic solutions constructed. We also present an algorithm for constructing all terms of the asymptotics of the logarithmic limit (logarithmic asymptotics) of the fundamental solution as t → +0 uniformly with respect to h. We write formulas of the asymptotics of the logarithmic limit for some special cases as t → +0. The method presented in this paper also allows us to construct exact asymptotics of solutions of initial–boundary value problems that are of probability meaning.     

Exact asymptotics in eigenproblems for fractional Brownian covariance operators

Many results in the theory of Gaussian processes rely on the eigenstructure of the covariance operator. However, eigenproblems are notoriously hard to solve explicitly and closed form solutions are known only in a limited number of cases. In this paper we set up a framework for the spectral analysis of the fractional type covariance operators, corresponding to an important family of processes, which includes the fractional Brownian motion and its noise. We obtain accurate asymptotic approximations for the eigenvalues and the eigenfunctions. Our results provide a key to several problems, whose solution is long known in the standard Brownian case, but was missing in the more general fractional setting. This includes computation of the exact limits of $L^2$-small ball probabilities and asymptotic analysis of singularly perturbed integral equations, arising in mathematical physics and applied probability.     

Existence and uniqueness of the solution,separation for certain second order elliptic differential equation

In this paper, we study conditions under which Schrodinger type operators with a matrix potential is separated and Schrodinger equation has a unique solution in the weighted space L_2,k(Rⁿ)^l, where l is any natural number and k ε C¹(Rⁿ) is a positive function     

Limit Theorems for the Non-linear Functional of Stationary Gaussian Processes

In this paper we consider two functional limit theorems for the non-linear functional of the stationary Gaussian process satisfying short range dependence conditions: the functional CLT for partial sum processes and the uniform CLT for a special class of functions. To carry out the proofs, we develop Rosenthal type inequalities for the functional of Gaussian processes.     

Small Ball Probabilities Around Random Centers of Gaussian Measures and Applications to Quantization

Let be a centered Gaussian measure on a separable Hilbert space (E, ). We are concerned with the logarithmic small ball probabilities around a -distributed center X. It turns out that the asymptotic behavior of –log (B(X,)) is a.s. equivalent to that of a deterministic function _R (). These new insights will be used to derive the precise asymptotics of a random quantization problem which was introduced in a former article by Dereich, Fehringer, Matoussi, and Scheutzow.⁽⁸⁾     

Asymptotics and Estimates of the Convergence Rate in a Three-Dimensional Boundary-Value Problem with Rapidly Alternating Boundary Conditions

We consider a singularly perturbed boundary-value eigenvalue problem for the Laplace operator in a cylinder with rapidly alternating type of the boundary condition on the lateral surface. The change of the boundary conditions is realized by splitting the lateral surface into many narrow strips on which the Dirichlet and Neumann conditions alternate. We study the case in which the averaged problem contains the Dirichlet boundary condition on the lateral surface. In the case of strips with slowly varying width we construct the first terms of the asymptotic expansions of eigenfunctions; moreover, in the case of strips with rapidly varying width we obtain estimates for the convergence rate.     

Sharp Small Deviation Asymptotics in the L 2-Norm for a Class of Gaussian Processes

We characterize the exact behavior of small deviations in a Hilbert norm for centered Gaussian processes in the case where their covariances have a special form of eigenvalues. This result enables us to describe small deviation asymptotics for certain Gaussian processes. Bibliography: 20 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 311, 2004, pp. 214–221.     

Asymptotic behavior of the distortion-rate function for Gaussian processes in Banach spaces

Let μ be a Gaussian measure on a separable Banach space. We prove a tight link between the logarithmic small ball probabilities of μ and certain moment generating functions. Based upon this link we provide a new lower bound for the distortion-rate function (DRF) against the small ball function. This allows us to use results of the theory of small ball probabilities to deduce lower bounds for the DRF. In particular, we obtain the correct weak asymptotics of the distortion rate function in many important cases (e.g. Brownian motion).     

On the Link Between Small Ball Probabilities and the Quantization Problem for Gaussian Measures on Banach Spaces

Let be a centered Gaussian measure on a separable Banach space E and N a positive integer. We study the asymptotics as N of the quantization error, i.e., the infimum over all subsets of E of cardinality N of the average distance w.r.t. to the closest point in the set . We compare the quantization error with the average distance which is obtained when the set is chosen by taking N i.i.d. copies of random elements with law . Our approach is based on the study of the asymptotics of the measure of a small ball around 0. Under slight conditions on the regular variation of the small ball function, we get upper and lower bounds of the deterministic and random quantization error and are able to show that both are of the same order. Our conditions are typically satisfied in case the Banach space is infinite dimensional.     

八阶奇异边值问题精确解的表达形式

该文在再生核空间W_2~9[0,1]中给出了求解八阶奇异边值问题的新算法.方程的精确解以级数形式给出.算例及数值结果验证了方法的实用性和有效性.     

Fast and Exact Simulation of Complex-Valued Stationary Gaussian Processes Through Embedding Circulant Matrix

This article is concerned with the study of the embedding circulant matrix method to simulate stationary complex-valued Gaussian sequences. The method is, in particular, shown to be well-suited to generate circularly symmetric stationary Gaussian processes. We provide simple conditions on the complex covariance function ensuring the theoretical validity of the minimal embedding circulant matrix method. We show that these conditions are satisfied by many examples and illustrate the simulation algorithm. In particular, we present a simulation study involving the circularly symmetric fractional Gaussian noise, a model introduced in this article. Supplementary material for this article is available online.     

Central Limit Theorems for Level Functionals of Stationary Gaussian Processes and Fields

We introduce a general method, which combines the one developed by authors in 1997 and one derived from the work of Malevich,⁽¹⁷⁾ Cuzick⁽⁷⁾ and mainly Berman,⁽³⁾ to provide in an easy way a CLT for level functionals of a Gaussian process, as well as a CLT for the length of a level curve of a Gaussian field.     

Small and Large Scale Asymptotics of some Lévy Stochastic Integrals

We provide general conditions for normalized, time-scaled stochastic integrals of independently scattered, Lévy random measures to converge to a limit. These integrals appear in many applied problems, for example, in connection to models for Internet traffic, where both large scale and small scale asymptotics are considered. Our result is a handy tool for checking such convergence. Numerous examples are provided as illustration. Somewhat surprisingly, there are examples where rescaling towards large times scales yields a Gaussian limit and where rescaling towards small time scales yields an infinite variance stable limit, and there are examples where the opposite occurs: a Gaussian limit appears when one converges towards small time scales and an infinite variance stable limit occurs when one converges towards large time scales.