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Exact asymptotics in eigenproblems for fractional Brownian covariance operators

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  • Chigansky, Pavel
  • Kleptsyna, Marina

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.

Suggested Citation

  • Chigansky, Pavel & Kleptsyna, Marina, 2018. "Exact asymptotics in eigenproblems for fractional Brownian covariance operators," Stochastic Processes and their Applications, Elsevier, vol. 128(6), pages 2007-2059.
    • Handle: RePEc:eee:spapps:v:128:y:2018:i:6:p:2007-2059
    DOI: 10.1016/j.spa.2017.08.019
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    1. van Zanten, Harry, 2007. "When is a linear combination of independent fBm's equivalent to a single fBm?," Stochastic Processes and their Applications, Elsevier, vol. 117(1), pages 57-70, January.
    2. Christian Bender & Tommi Sottinen & Esko Valkeila, 2010. "Fractional processes as models in stochastic finance," Papers 1004.3106, arXiv.org.
    3. Le Breton, Alain, 1998. "Filtering and parameter estimation in a simple linear system driven by a fractional Brownian motion," Statistics & Probability Letters, Elsevier, vol. 38(3), pages 263-274, June.
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    Cited by:

    1. Nazarov, A.I. & Nikitin, Ya.Yu., 2021. "Gaussian processes centered at their online average, and applications," Statistics & Probability Letters, Elsevier, vol. 170(C).

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