Exact asymptotics in eigenproblems for fractional Brownian covariance operators

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 L 2 -small ball probabilities and asymptotic analysis of singularly perturbed integral equations, arising in mathematical physics and applied probability.

Mixed fractional Brownian motion: A spectral take

“…This lemma is checked exactly as in [9], see calculations preceding (5.39) therein. At this stage, the eigenproblem (2.1) is equivalent to finding all ν > 0, for which there exists a nonzero vector C 1 C 2 satisfying (3.6) with Φ defined by (3.11).…”

“…Implementation of this program uses the technique of solving the Riemann boundary value problem, see [13]. We will detail its main steps, referring the reader to the relevant parts in [9], whenever calculations are similar.…”

“…(5.35)] and consequently Φ 0 (z) and Φ 1 (z), satisfying the conditions (3.3)-(3.4), can be expressed in terms of solutions to certain integral equations, see (3.10) below. More precisely, define sectionally holomorphic function 9) and the real valued function…”

“…The limiting performance of statistical procedures in models involving mixed fBm is governed by the asymptotic behaviour of this equation as t → ∞, see [8]. For H > 1 2 , under reparametrization ε := t 1−2H it reduces to the singularly perturbed problem( 1.2) As ε → 0 its solution g ε (x) converges to the solution g 0 (x) of the limit equation, obtained by setting ε := 0 in (1.2), with the following rate with respect to L 2 -norm, see [9],…”

“…( 1.2) As ε → 0 its solution g ε (x) converges to the solution g 0 (x) of the limit equation, obtained by setting ε := 0 in (1.2), with the following rate with respect to L 2 -norm, see [9],…”