Generalized random fields and Lévy's continuity Theorem on the space of tempered distributions

Hermine Biermé (1) , Olivier Durieu (2, 3) , Yizao Wang (4)

1 LMA [Poitiers] - Laboratoire de mathématiques et applications [UMR 7348]
2 FDP - Fédération de recherche Denis Poisson
3 LMPT - Laboratoire de Mathématiques et Physique Théorique
4 Department of Mathematical Sciences [Cincinnati]

In this note, we recall main properties of generalized random fields and present a proof of the continuity theorem of Paul Lévy for generalized random fields in the space of tempered distributions. This theorem was first proved by Fernique in a more general setting. The aim of this note is to provide a self-contained proof that in particular avoids the abstract theory of nuclear spaces.

Mots clés: Generalized random fields; tempered distributions; Lévy's continuity theorem; Minlos--Bochner's theorem

Domaines: Probabilités [math.PR]

Hermine Biermé, Olivier Durieu, Yizao Wang. Generalized random fields and Lévy's continuity Theorem on the space of tempered distributions. 2017. ⟨hal-01568486⟩