Exact asymptotics in eigenproblems for fractional Brownian covariance operators

Pavel Chigansky, Marina Kleptsyna*

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

17 Scopus citations

Abstract

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 L2-small ball probabilities and asymptotic analysis of singularly perturbed integral equations, arising in mathematical physics and applied probability.

Original languageAmerican English
Pages (from-to)2007-2059
Number of pages53
JournalStochastic Processes and their Applications
Volume128
Issue number6
DOIs
StatePublished - Jun 2018

Bibliographical note

Publisher Copyright:
© 2017 Elsevier B.V.

Keywords

  • Eigenproblem
  • Fractional Brownian motion
  • Gaussian processes
  • Karhunen–Loève expansion
  • Optimal linear filtering
  • Small ball probabilities
  • Spectral asymptotics

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