A generalization of the contraction principle in the space of convergent sequences. (English. Russian original) Zbl 0738.60029

Buldygin, V. V.; Solntsev, S. A.

Theory Probab. Math. Stat. 43, 35-39 (1991); translation from Teor. Veroyatn. Mat. Stat., Kiev 43, 32-37 (1990).

Summary: Let \((Y_{nk};\;n,k\geq 1)\) be an array of random elements of a separable Banach space \({\mathfrak X};(\beta_{nk}; n,k\geq 1)\) is a real contraction array; \(c({\mathfrak X})\) is the space of sequences convergent in the norm of \({\mathfrak X}\). Conditions on \((Y_{nk})\) and \((\beta_{nk})\) are established under which the condition that \((\sum^ \infty_{k=1}Y_{nk},\;n\geq 1)\in c({\mathfrak X})\) a.s. implies that \((\sum^ \infty_{k=1}\beta_{nk}Y_{nk},\;n\geq 1)\in c({\mathfrak X})\) a.s.

MSC:

60G17	Sample path properties
46B25	Classical Banach spaces in the general theory

Keywords:

contraction principle; array of random elements; Banach space