English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Zendy Open

Presents the pdf analysis as premium feature

PDF Analysis

Insights

Presents the summarisation as premium feature

Summarisation

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the zaia usage as premium feature

ZAIA

Zendy Tools

Zendy Plus

Presents the access of premium content as premium feature

Premium Content

Many results in the theory of Gaussian processes rely on the eigenstructureof the covariance operator. However, eigenproblems are notoriously hard tosolve explicitly and closed form solutions are known only in a limited numberof cases. In this paper we set up a framework for the spectral analysis of thefractional type covariance operators, corresponding to an important family ofprocesses, which includes the fractional Brownian motion and its noise. Weobtain accurate asymptotic approximations for the eigenvalues and theeigenfunctions. Our results provide a key to several problems, whose solutionis long known in the standard Brownian case, but was missing in the moregeneral fractional setting. This includes computation of the exact limits of$L^2$-small ball probabilities and asymptotic analysis of singularly perturbedintegral equations, arising in mathematical physics and applied probability.

Exact asymptotics in eigenproblems for fractional Brownian covariance operators