On the law of large numbers in 2-uniformly smooth Banach spaces
Kolmogorov's strong law of large numbers is extended to random variables which take their values in a 2-uniformly smooth Banach space \((B,\| \cdot \|)\). The result reads as follows: Let \((X_ k\), \(k\in {\mathbb{N}})\) be a sequence of centered independent random variables taking values in \((B,\| \cdot \|)\). Suppose that the following conditions hold: 1) There exists a positive constant k such that: \(\| X_ n\| \leq kn/\sqrt{L_ 2n}^ a.\)s., all \(n\in {\mathbb{N}}\), where \(L_ 2n=\log(\log(\sup \{n,e^ e\}));\quad 2)\quad \lim_{n\to \infty}(\sum^{n}_{j=1}E\| X_ j\|^ 2)/n^ 2=0;\quad 3)\quad \sum^{\infty}_{j=1}\sup_{\| f\|_{B'}\leq 1}(Ef^ 2(X_ j))/j^ 2<+\infty.\) Then the sequence \((X_ k\), \(k\in {\mathbb{N}})\) satisfies the strong law of large numbers.
scientific article; zbMATH DE number 3868316