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Small ball probabilities around random centers of Gaussian measures and applications to quantization

TitleSmall ball probabilities around random centers of Gaussian measures and applications to quantization
Publication TypeJournal Article
Year of Publication2003
AuthorsSteffen Dereich
JournalJ. Theor. Probab.
Volume16(2)
Pagination427-449
KeywordsGaussian process, quantization, small ball probabilites for random centers
Abstract

Let $ \mu $ be a centered Gaussian measure on a separable Hilbert space $ (E, \| \cdot \|) $. We are concerned with the logarithmic small ball probabilities around a $ \mu $-distributed center $ X $. It turns out that the asymptotic behavior of $ –\log \ \mu(B(X,\epsilon)) $ is a.s. equivalent to that of a deterministic function $ \phi_R (\epsilon) $. These new insights will be used to derive the precise asymptotics of a random quantization problem which was introduced in a former article by Dereich, Fehringer, Matoussi, and Scheutzow [8].