相关论文: Degenerate self-similar measures, spectral asympto…

We discuss the centering operation for the Green Gaussian processes and calculate $L_2$-small ball asymptotics for some centered (demeaned) processes.

In this article we study the small ball probabilities in $L_2$-norm for a family of finite-dimensional perturbations of Gaussian functions. We define three types of perturbations: non-critical, partially critical and critical; and derive…

We prove a new variant of comparison principle for logarithmic $L_2$-small ball probabilities of Gaussian processes. As an application, we obtain logarithmic small ball asymptotics for some well-known processes with smooth covariances.

We sharpen a classical result on the spectral asymptotics of the boundary value problems for self-adjoint ordinary differential operator. Using this result we obtain the exact $L_2$-small ball asymptotics for a new class of zero mean…

This article is a survey of the results on asymptotic behavior of small ball probabilities in $L_2$-norm. Recent progress in this field is mainly based on the methods of spectral theory of differential and integral operators.

We find the logarithmic $L_2$-small ball asymptotics for a class of zero mean Gaussian fields with covariances having the structure of "tensor product". The main condition imposed on marginal covariances is slow growth at the origin of…

We study the small ball asymptotics problem in $L_2$ for two generalizations of the fractional Brownian motion with variable Hurst parameter. To this end, we perform careful analysis of the singular values asymptotics for associated…

We study spectral problems for integro-differential equations arising in the theory of Gaussian processes similar to the fractional Brownian motion. We generalize the method of Chigansky--Kleptsyna and obtain the two-term eigenvalue…

We find exact small deviation asymptotics with respect to weighted Hilbert norm for some well-known Gaussian processes. Our approach does not require the knowledge of eigenfunctions of the covariance operator of a weighted process. Such a…

We prove comparison theorems for small ball probabilities of the Green Gaussian processes in weighted $L_2$-norms. We find the sharp small ball asymptotics for many classical processes under quite general assumptions on the weight.

The sharp asymptotics for the L^2-quantization errors of Gaussian measures on a Hilbert space and, in particular, for Gaussian processes is derived. The condition imposed is regular variation of the eigenvalues.

We consider a set of one-dimensional transformations of Gaussian random functions. Under natural assumptions we obtain a connection between $L_2$-small ball asymptotics of the transformed function and of the original one. Also the explicit…

We find logarithmic asymptotics of $L_2$-small deviation probabilities for weighted stationary Gaussian processes (both for real and complex-valued) having power-type discrete or continuous spectrum. As in the recent work by Hong, Lifshits…

While small ball, or lower tail, asymptotic for Gaussian measures generated by solutions of stochastic ordinary differential equations is relatively well understood, a lot less is known in the case of stochastic partial differential…

We investigate the asymptotic properties of the minimum $L_1$-norm estimator of the drift parameter for fractional Ornstein-Uhlenbeck type process driven by a general Gaussian process.

Eigenproblems frequently arise in theory and applications of stochastic processes, but only a few have explicit solutions. Those which do, are usually solved by reduction to the generalized Sturm--Liouville theory for differential…

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.…

Starting from the notion of multivariate fractional Brownian Motion introduced in [F. Lavancier, A. Philippe, and D. Surgailis. Covariance function of vector self-similar processes. Statistics & Probability Letters, 2009] we define a…

The paper is devoted to three-parametric self-similar Gaussian Volterra processes that generalize fractional Brownian motion. We study the asymptotic growth of such processes and the properties of long- and short-range dependence. Then we…

These are lecture notes from a course given at the CRM in Montreal in 1992. They survey the author's attempts to find and understand canonical probabilistic entities in a local field (e.g. p-adic) setting. We propose answers to the related…