On the convergence of certain sums of independent random elements. (English) Zbl 1090.46009

Ferrando, J. C.

Commentat. Math. Univ. Carol. 43, No. 1, 77-81 (2002).

Summary: In this note, we investigate the relationship between the convergence of the sequence \(\{S_{n}\}\) of sums of independent random elements of the form \(S_{n}=\sum _{i=1}^{n}\varepsilon _{i}x_{i}\) (where \(\varepsilon _{i}\) takes the values \(\pm 1\) with the same probability and \(x_{i}\) belongs to a real Banach space \(X\) for each \(i\in \mathbb N\)) and the existence of certain weakly unconditionally Cauchy subseries of \(\sum _{n=1}^{\infty }x_{n}\).

MSC:

46B15	Summability and bases; functional analytic aspects of frames in Banach and Hilbert spaces
46B09	Probabilistic methods in Banach space theory

Keywords:

independent random elements; copy of \(c_{0}\); Pettis integrable function; perfect measure space