On the exponential integrability of quasi-additive functionals of Gaussian vectors. (Ukrainian, English) Zbl 1050.60005

Buldygin, V. V.

Teor. Jmovirn. Mat. Stat. 68, 18-23 (2003); translation in Theory Probab. Math. Stat. 68, 19-26 (2004).

The main result of this paper is the following: Let \((V,{\mathcal F})\) be a measurable vector space, let \(X\) be a \(V\)-valued Gaussian random vector, let \(C\geq1\), and let \(g(\cdot)\) be a measurable \(C\)-quasi-additive functional on \(V\). Then for any \(\varepsilon>0\) there exists a number \(a_{0}>0\), such that for all \(a\in[0,a_0)\): \(E\exp\{ag^{b}(X)\}<\infty\), where \(b=b(\varepsilon,C)=2[1+(1+\varepsilon)\log_{2}C]^{-1}\). If, in addition, \(C\in[1,2)\) and \(g(\cdot)\) is convex, then there exists a number \(a_0>0\), such that for all \(a\in[0,a_0)\): \(E\exp\{ag^{b}(X)\}<\infty\), where \(b=2[1+\log_2 C]^{-1}\).

Reviewer: A. D. Borisenko (Kyïv)

MSC:

60B11	Probability theory on linear topological spaces
60G15	Gaussian processes

Keywords:

exponential integrability; quasi-additive functionals; Gaussian vector