I: 05, 54-71, LNM 39 (1967)
FERNIQUE, Xavier
Séries de distributions aléatoires indépendantes (2 talks) (
Miscellanea)
This is part of X.~Fernique's research on random distributions (probability measures on ${\cal D}'$, and more generally on the dual space $E'$ of a nuclear LF space $E$) and their characteristic functions, which are exactly, according to Minlos' theorem, the continuous positive definite functions on $E$ assuming the value $1$ at $0$. Here it is proved that a series of independent random distributions converges a.s. if and only if the product of their characteristic functions converges pointwise to a continuous limit, and converges a.s. after centering if and only if the product of absolute values converges
Comment: See for further results
Ann. Inst. Fourier, 17-1, 1967;
Invent. Math.,
3, 1967, and
C.R. Acad. Sc.,
266, 1968 for the extension of Lévy's continuity theorem (also presented at Séminaire Bourbaki, June 1966,
311)
Keywords: Random distributions,
Minlos theoremNature: Original
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