Estimates of moments of supremum of normed sums of independent random variables. (Ukrainian, English) Zbl 1052.60035

Matsak, I. K.

Teor. Jmovirn. Mat. Stat. 67, 104-116 (2002); translation in Theory Probab. Math. Stat. 67, 115-128 (2003).

Let \((\xi_n)\) be a sequence of independent random variables, let \((a_n)\) be a nondecreasing real-valued sequence such that \(a_n \to \infty\) as \(n \to \infty\). Two-sided bounds for \(\sup_{n\geq 1} (a_n^{-1}| \sum_{i=1}^n \xi_i| )\) and for \(\sup_{n\geq 1} (a_n^{-1}| \xi_i| )\) are obtained. Extensions to the case of Banach spaces and Banach lattices are derived. Several examples of application of the obtained inequalities to prove limit theorems such as the law of large numbers and the law of the iterated logarithm are presented.

Reviewer: N. M. Zinchenko (Kyïv)

MSC:

60G50	Sums of independent random variables; random walks
60F15	Strong limit theorems
46B42	Banach lattices

Keywords:

moments; sums of random variables; supremum; Banach space; Banach lattice; Law of Large Numbers; Law of the Iterated Logarithm