Summary
In this paper we define two classes of Banach space (B, ∥·∥)-valued random vectors called sub-Gaussian vectors and γ-sub-Gaussian vectors. The main purpose of this paper is to prove the exponential integrability of a sub-Gaussian vectorX, that is,\(\mathbb{E}[e^{\varepsilon \parallel X\parallel 2} ]< \infty\) for some ε>0, in the case whereB=L p . On the other hand, using the arguments ofX. Fernique and M. Talagrand, we also show that the exponential integrability of a γ-sub-Gaussian vector in an arbitrary separable Banach space.
These two definitions of sub-Gaussian vectors and γ-sub-Gaussian vectors are not comparable, and neither of these definitions is a necessary condition for the exponential integrability. We shall give illuminating examples.
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Fukuda, R. Exponential integrability of sub-Gaussian vectors. Probab. Th. Rel. Fields 85, 505–521 (1990). https://doi.org/10.1007/BF01203168
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DOI: https://doi.org/10.1007/BF01203168