Exact asymptotics in eigenproblems for fractional Brownian covariance operators

STOCHASTIC PROCESSES AND THEIR APPLICATIONS

卷:128

Article

Chigansky, Pavel¹  Kleptsyna, Marina² 

[1] Hebrew Univ Jerusalem, Dept Stat, IL-91905 Jerusalem, Israel

[2] Univ Maine, Lab Stat & Proc, Le Mans, France

关键词: Gaussian processes;    Fractional Brownian motion;    Spectral asymptotics;    Eigenproblem;    Small ball probabilities;    Optimal linear filtering;    Karhunen-Loeve expansion;   

DOI  :  10.1016/j.spa.2017.08.019

来源: Elsevier

【 摘 要 】

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. (C) 2017 Elsevier B.V. All rights reserved.

【 授权许可】

Free