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 Mat. Sb., 1998, Volume 189, Number 5, Pages 47–68 (Mi msb319)

On the small balls problem for equivalent Gaussian measures

V. I. Bogachev

M. V. Lomonosov Moscow State University

Abstract: Let $\mu$ be a centred Gaussian measure in a linear space $X$ with Cameron-Martin space $H$, let $q$ be a $\mu$-measurable seminorm, and let $Q$ be a $\mu$-measurable second-order polynomial. We show that it is sufficient for the existence of the limit $\lim _{\varepsilon \to 0}\mathsf E(\exp Q|q\leqslant \varepsilon)$, where $E$ is the expectation with respect to $\mu$, that the second derivative $D_{H}^{ 2}Q$ of the function $Q$ be a nuclear operator on $H$. This condition is also necessary for the existence of the above-mentioned limit for all seminorms $q$. The problem under discussion can be reformulated as follows: study $\lim _{\varepsilon \to 0}\nu (q\leqslant \varepsilon )/\mu (q\leqslant \varepsilon )$ for Gaussian measures $\nu$ equivalent to $\mu$.

DOI: https://doi.org/10.4213/sm319

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English version:
Sbornik: Mathematics, 1998, 189:5, 683–705

Bibliographic databases:

UDC: 512.55
MSC: 28C20, 60B11

Citation: V. I. Bogachev, “On the small balls problem for equivalent Gaussian measures”, Mat. Sb., 189:5 (1998), 47–68; Sb. Math., 189:5 (1998), 683–705

Citation in format AMSBIB
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• http://mi.mathnet.ru/eng/msb319
• https://doi.org/10.4213/sm319
• http://mi.mathnet.ru/eng/msb/v189/i5/p47

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This publication is cited in the following articles:
1. V. R. Fatalov, “Constants in the asymptotics of small deviation probabilities for Gaussian processes and fields”, Russian Math. Surveys, 58:4 (2003), 725–772
2. Kara-Zaitri L., Laksaci A., Rachdi M., Vieu Ph., “Uniform in bandwidth consistency for various kernel estimators involving functional data”, J. Nonparametr. Stat., 29:1 (2017), 85–107
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