Small ball probabilities for Gaussian random fields and tensor products of compact operators

Abstract

Abstract. We find the logarithmic L 2 -small ball asymptotics for a large class of zero mean Gaussian fields with covariances having the structure of "tensor product". The main condition imposed on marginal covariances is the regular behavior of their eigenvalues at infinity that is valid for a multitude of Gaussian random functions including the fractional Brownian sheet, Ornstein -Uhlenbeck sheet, etc. So we get the far-reaching generalizations of well-known results by Csáki (1982) and by Li (1992). Another class of Gaussian fields considered is the class of additive fields studied under the supremum-norm by Chen and Li (2003). Our theorems are based on new results on spectral asymptotics for the tensor products of compact self-adjoint operators in Hilbert space which are of independent interest.

Approximation complexity of additive random fields

“…in Csáki [3], Li [7] Papageorgiou and Wasilkowski [10] (for q = 0) and especially in Karol' et al [5] for a case that is even more general than what we need here. Recently, N. Serdyukova (private communication) investigated some cases not covered by Assumption 4.1 by using the Mellin transform formalism from [5]. For example, for q = r > 1/2 (in other words, = −1) she obtained 2 n,h ≈ n −2r (log n) −2r (log log n) 2r(h−1) , which, of course, agrees with (4.1) up to the two main terms but exhibits an extra factor with iterated logarithm.…”

Lectures on Gaussian Processes

“…For example, very accurate calculations yield the following result [88]. For example, very accurate calculations yield the following result [88].…”

Approximation complexity of sums of random processes

“…, s d ) are from [0, 1] d . Such random fields belong to a wide class of so-called additive random fields (see [2] and [7]). Every Y d is considered as a random element of the space L 2 ([0, 1] d ) endowed with the scalar product · , · 2,d and norm · 2,d .…”