On the central limit theorem in Banach spaces

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Abstract

It is shown, with the use of a concentration inequality of LeCam, that associated with an infinitely divisible random variable with values in a separable Banach space there is a Lévy-Khintchine formula. A partial converse of this fact is also proved. Relations between the continuity of the compound Poisson and the Gaussian variables associated with a Lévy measure are studied. A central limit theorem is obtained and examples are given.

MSC

60F05

MSC

60G99

Keywords

Central limit theorem in Banach spaces
LeCam concentration inequality
Lévy-Khintchine representation in Banach spaces

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This research was supported in part by the U.S. Army Research Office, Durham, under Grant DA-ARO-D-31-124-73-G31, and in part by National Science Foundation under Grant GP-31091X.