The order law of large numbers in Banach lattices. (English. Ukrainian original) Zbl 1005.60015

Matsak, I. K.

Theory Probab. Math. Stat. 62, 89-102 (2001); translation from Teor. Jmovirn. Mat. Stat. 62, 83-95 (2000).

Let \(B\) be a Banach lattice and let \(X_{i}\), \(i\geq 1\), be a sequence of independent random elements with values in \(B,\) and let \(S_{n}=\sum_{i=1}^{n}X_{i}.\) It is known that the sequence \(X_{i},\) \(EX_{i}=0,\) satisfies the law of large numbers if \(\lim\frac{S_{n}}{n}=0\) a.s., where the limit is understood in the sense of modulo of the space \(B.\) In the present paper, conditions under which the latter relation is fulfilled are proposed.

Reviewer: A.V.Swishchuk (Kyïv)

MSC:

60B12	Limit theorems for vector-valued random variables (infinite-dimensional case)
60G50	Sums of independent random variables; random walks
60F15	Strong limit theorems

Keywords:

Banach lattice; law of large numbers