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

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Small ball probabilities for smooth Gaussian fields and tensor products of compact operators. / Karol', Andrei I.; Nazarov, Alexander I.

In: Mathematische Nachrichten, Vol. 287, No. 5-6, 2014, p. 595-609.

Research output: Contribution to journal › Article › peer-review

BibTeX

@article{313bd07b5bea44b09047b5dd5d0324a8,

title = "Small ball probabilities for smooth Gaussian fields and tensor products of compact operators",

abstract = "We find the logarithmic L-2-small ball asymptotics for a class of zero mean Gaussian fields with covariances having the structure of tensor product. The main condition imposed on marginal covariances is slow growth at the origin of counting functions of their eigenvalues. That is valid for Gaussian functions with smooth covariances. Another type of marginal functions considered as well are classical Wiener process, Brownian bridge, Ornstein-Uhlenbeck process, etc., in the case of special self-similar measure of integration. Our results are based on a new theorem on spectral asymptotics for the tensor products of compact self-adjoint operators in Hilbert space which is of independent interest. Thus, we continue to develop the approach proposed in the paper , where the regular behavior at infinity of marginal eigenvalues was assumed.",

author = "Karol', {Andrei I.} and Nazarov, {Alexander I.}",

year = "2014",

doi = "DOI 10.1002/mana.201100010",

language = "English",

volume = "287",

pages = "595--609",

journal = "Mathematische Nachrichten",

issn = "0025-584X",

publisher = "Wiley-Blackwell",

number = "5-6",

}

RIS

TY - JOUR

T1 - Small ball probabilities for smooth Gaussian fields and tensor products of compact operators

AU - Karol', Andrei I.

AU - Nazarov, Alexander I.

PY - 2014

Y1 - 2014

N2 - We find the logarithmic L-2-small ball asymptotics for a class of zero mean Gaussian fields with covariances having the structure of tensor product. The main condition imposed on marginal covariances is slow growth at the origin of counting functions of their eigenvalues. That is valid for Gaussian functions with smooth covariances. Another type of marginal functions considered as well are classical Wiener process, Brownian bridge, Ornstein-Uhlenbeck process, etc., in the case of special self-similar measure of integration. Our results are based on a new theorem on spectral asymptotics for the tensor products of compact self-adjoint operators in Hilbert space which is of independent interest. Thus, we continue to develop the approach proposed in the paper , where the regular behavior at infinity of marginal eigenvalues was assumed.

AB - We find the logarithmic L-2-small ball asymptotics for a class of zero mean Gaussian fields with covariances having the structure of tensor product. The main condition imposed on marginal covariances is slow growth at the origin of counting functions of their eigenvalues. That is valid for Gaussian functions with smooth covariances. Another type of marginal functions considered as well are classical Wiener process, Brownian bridge, Ornstein-Uhlenbeck process, etc., in the case of special self-similar measure of integration. Our results are based on a new theorem on spectral asymptotics for the tensor products of compact self-adjoint operators in Hilbert space which is of independent interest. Thus, we continue to develop the approach proposed in the paper , where the regular behavior at infinity of marginal eigenvalues was assumed.

U2 - DOI 10.1002/mana.201100010

DO - DOI 10.1002/mana.201100010

M3 - Article

VL - 287

SP - 595

EP - 609

JO - Mathematische Nachrichten

JF - Mathematische Nachrichten

SN - 0025-584X

IS - 5-6

ER -

ID: 5667602