La loi des grands nombres de Prokhorov dans les espaces de type p. (The Prokhorov law of large numbers for spaces of type p) (Q1103253)

The main result of the paper is the extension of Prokhorov's law of large numbers to random elements \(X_ 1\), \(X_ 2\),... of a separable Banach space B of type p, \(1<p\leq 2\). The classical assumption of the scalar case is replaced by the following one: \[ \forall \epsilon >0: \sum^{\infty}_{n=1}\exp \{-\epsilon \lambda_ n^{- 1}\}<\infty,\text{ where } \lambda_ n=2^{-2n}\sup \{\sum^{2^{n+1}}_{i=2^ n+1}E <x^*, X_ i>^ 2;\;\| x^*\| \leq 1\}. \] The necessity of this condition is investigated.