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Small Deviations of Gaussian Random Fields in $L_q$-Spaces

Dublin Core		PKP Metadata Items	Metadata for this Document
1.	Title	Title of document	Small Deviations of Gaussian Random Fields in $L_q$-Spaces
2.	Creator	Author's name, affiliation, country	Mikhail Lifshits; St.Petersburg State University
2.	Creator	Author's name, affiliation, country	Werner Linde; FSU Jena
2.	Creator	Author's name, affiliation, country	Zhan Shi; Universite Paris VI
3.	Subject	Discipline(s)
3.	Subject	Keyword(s)	Gaussian random fields; fractional Brownian motion; fractal measures; Kolmogorov numbers; metric entropy
3.	Subject	Subject classification	60G15; 28A80; 47B06
4.	Description	Abstract	We investigate small deviation properties of Gaussian random fields in the space $L_q(R^N,\mu)$ where $\mu$ is an arbitrary finite compactly supported Borel measure. Of special interest are hereby "thin" measures $\mu$, i.e., those which are singular with respect to the $N$--dimensional Lebesgue measure; the so-called self-similar measures providing a class of typical examples. For a large class of random fields (including, among others, fractional Brownian motions), we describe the behavior of small deviation probabilities via numerical characteristics of $\mu$, called mixed entropy, characterizing size and regularity of $\mu$. For the particularly interesting case of self-similar measures $\mu$, the asymptotic behavior of the mixed entropy is evaluated explicitly. As a consequence, we get the asymptotic of the small deviation for $N$-parameter fractional Brownian motions with respect to $L_q(R^N,\mu)$-norms. While the upper estimates for the small deviation probabilities are proved by purely probabilistic methods, the lower bounds are established by analytic tools concerning Kolmogorov and entropy numbers of Holder operators.
5.	Publisher	Organizing agency, location
6.	Contributor	Sponsor(s)
7.	Date	(YYYY-MM-DD)	2006-12-08
8.	Type	Status & genre	Peer-reviewed Article
8.	Type	Type
9.	Format	File format	PDF
10.	Identifier	Uniform Resource Identifier	http://ejp.ejpecp.org/article/view/379
10.	Identifier	Digital Object Identifier	10.1214/EJP.v11-379
11.	Source	Journal/conference title; vol., no. (year)	Electronic Journal of Probability; Vol 11
12.	Language	English=en
14.	Coverage	Geo-spatial location, chronological period, research sample (gender, age, etc.)
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