Ball Asymptotics Research Articles

Formula omitted]-small ball asymptotics for some demeaned Gaussian processes

We discuss the centering operation for the Green Gaussian processes and calculate L2-small ball asymptotics for some centered (demeaned) processes.

L2-Small Ball Asymptotics for a Family of Finite-Dimensional Perturbations of Gaussian Functions

L2-Small Ball Asymptotics for a Family of Finite-Dimensional Perturbations of Gaussian Functions

Spectral Analysis for Some Multifractional Gaussian Processes

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 integral operators.

Exact L2-Small Ball Asymptotics for Some Durbin Processes

Exact L2-Small Ball Asymptotics for Some Durbin Processes

Comparison theorems for the small ball probabilities of the Green Gaussian processes in weighted $L_2$-norms

Comparison theorems are proved for small ball probabilities of the Green Gaussian processes in weighted $L_2$-norms. The sharp small ball asymptotics are found for many classical processes under quite general assumptions on the weight.

Small ball probabilities for smooth Gaussian fields and tensor products of compact operators

We find the logarithmic L2‐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 counting functions of their eigenvalues. That is valid for Gaussian functions with smooth covariances. Another type of marginal functions considered as well are classical Wiener process, Brownian bridge, Ornstein–Uhlenbeck process, etc., in the case of special self‐similar measure of integration. Our results are based on a new theorem on spectral asymptotics for the tensor products of compact self‐adjoint operators in Hilbert space which is of independent interest. Thus, we continue to develop the approach proposed in the paper , where the regular behavior at infinity of marginal eigenvalues was assumed.

Degenerate self-similar measures, spectral asymptotics and small deviations of Gaussian processes

We find the logarithmic small ball asymptotics for the L2-norm with respect to degenerate self-similar measures of a certain class of Gaussian processes, including Brownian motion, Ornstein–Uhlenbeck process and their integrated counterparts.

Exact small ball asymptotics in weighted L 2-norm for some Gaussian processes

We find the exact small ball asymptotics under weighted L 2-norm for a wide class of Gaussian processes which generate boundary-value problems for ordinary differential equations. Sharp constants in the asymptotics are derived for a number of processes connected with special functions. Bibliography: 23 titles.

Log-level comparison principle for small ball probabilities

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.

Exact L 2-Small Ball Asymptotics of Gaussian Processes and the Spectrum of Boundary-Value Problems

We sharpen a classical result on the spectral asymptotics of boundary-value problems for self-adjoint ordinary differential operator. Using this result, we obtain the exact L2-small ball asymptotics for a new class of zero-mean Gaussian processes. This class includes, in particular, the integrated generalized Slepian process, integrated centered Wiener process, and integrated centered Brownian bridge.

Small ball probabilities for Gaussian random fields and tensor products of compact operators

We find the logarithmic L 2 -small ball asymptotics for a large class of zero mean Gaussian fields with covariances having the structure of product. The main condition imposed on marginal covariances is the regular behavior of their eigenvalues at infinity that is valid for a multitude of Gaussian random functions including the fractional Brownian sheet, Ornstein - Uhlenbeck sheet, etc. So we get the far-reaching generalizations of well-known results by Csaki (1982) and by Li (1992). Another class of Gaussian fields considered is the class of additive fields studied under the supremum-norm by Chen and Li (2003). Our theorems are based on new results on spectral asymptotics for the tensor products of compact self-adjoint operators in Hilbert space which are of independent interest.

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.

Logarithmic L2-Small Ball Asymptotics for some Fractional Gaussian Processes

We find the logarithmic L2-small ball asymptotics of some Gaussian processes related to the fractional Brownian motion (fBm), fractional Ornstein--Uhlenbeck process (fOU), and their integrated analogues. We consider also the multiparameter generalizations.

Small Ball Asymptotics for the Stochastic Wave Equation

We examine the small ball asymptotics for the weak solution X of the stochastic wave equation $$\frac{{\partial ^2 X}}{{\partial t^2 }}(t,x) - \frac{{\partial ^2 X}}{{\partial x^2 }}(t,x) = g(X(t,x)) + f(t,x)dW(t,x)$$ on the real line with deterministic initial conditions.

Exact L 2 -small ball behavior of integrated Gaussian processes and spectral asymptotics of boundary value problems

We find the exact small deviation asymptotics for the L2-norm of various m-times integrated Gaussian processes closely connected with the Wiener process and the Ornstein – Uhlenbeck process. Using a general approach from the spectral theory of linear differential operators we obtain the two-term spectral asymptotics of eigenvalues in corresponding boundary value problems. This enables us to improve the recent results from [15] on the small ball asymptotics for a class of m-times integrated Wiener processes. Moreover, the exact small ball asymptotics for the m-times integrated Brownian bridge, the m-times integrated Ornstein – Uhlenbeck process and similar processes appear as relatively simple examples illustrating the developed general theory.

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 L2-Norm

We find the sharp constant in the small L2-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.