Previous |  Up |  Next


independent random elements; copy of $c_{0}$; Pettis integrable function; perfect measure space
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 $i\in \Bbb N$) and the existence of certain weakly unconditionally Cauchy subseries of $\sum_{n=1}^{\infty}x_{n}$.
[1] Cembranos P., Mendoza J.: Banach Spaces of Vector-Valued Functions. LNM 1676, Springer, 1997. MR 1489231 | Zbl 0902.46017
[2] Díaz S., Fernández A., Florencio M., Paúl P.J.: Complemented copies of $c_{0}$ in the space of Pettis integrable functions. Quaestiones Math. 16 (1993), 61-66. MR 1217475
[3] Diestel J.: Sequences and series in Banach spaces. GTM 92, Springer-Verlag, New York-Berlin-Heidelberg-Tokyo, 1984. MR 0737004
[4] Diestel J., Uhl J.: Vector measures. Math Surveys 15, Amer. Math. Soc., Providence, 1977. MR 0453964 | Zbl 0521.46035
[5] Ferrando J.C.: On a theorem of Kwapień. Quaestiones Math. 24 (2001), 51-54. MR 1824912 | Zbl 1019.46010
[6] Freniche F.J.: Embedding $c_{0}$ in the space of Pettis integrable functions. Quaestiones Math. 21 (1998), 261-267. MR 1701785 | Zbl 0963.46025
[7] Halmos P.R.: Measure Theory. GTM 18, Springer, New York-Berlin-Heidelberg-Barcelona, 1950. MR 0033869 | Zbl 0283.28001
[8] Kwapień S.: On Banach spaces containing $c_{0}$. Studia Math. 52 (1974), 187-188. MR 0356156
[9] Vakhania N.N., Tarieladze V.I., Chobanian S.A.: Probability Distributions on Banach Spaces. D. Reidel Publishing Company, Dordrecht, 1987. MR 1435288
Partner of
EuDML logo