The order law of large numbers in Banach lattices (Q2737028)

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.