Prokhorov-Loeve strong law of large numbers for martingales normalized by operators. (Ukrainian, English) Zbl 1119.60022

Buldygin, V. V.; Koval’, V. O.

Teor. Jmovirn. Mat. Stat. 73, 27-42 (2005); translation in Theory Probab. Math. Stat. 73, 31-46 (2006).

Let \(\{X_n,n\geq1\}\) be a sequence of independent symmetric random vectors in \(\mathbb R^m\), let \(\{A_n,n\geq1\}\) be a sequence of non-random linear operators \(A_N\in {\mathcal L}(\mathbb R^m,\mathbb R^d)\), let \(\mathcal N\) be the set of all monotonically increasing to infinity sequences of natural numbers, and let \(S_n=\sum_{i=1}^{n}X_i\). The Prokhorov-Loeve strong law of large numbers gives conditions under which the convergence \(\| A_nS_n\| \to0\) a.e. as \(n\to\infty\) is equivalent to convergence \(\| A_{n_{j+1}}(S_{n_{j+1}}-S_{n_j})\| \to0\) a.e. as \(j\to\infty\) for sequences \(\{n_j,j\geq1\}\) from a subset \(\tilde{\mathcal N}={\mathcal N}(\{A_n\})\) of the set \(\mathcal N\) that is called test class for the sequence \(\{A_n,n\geq1\}\). In the first part of the article the authors propose an algorithm for constructing the test classes random vectors normalized by linear operators in a finite-dimensional Euclidean space. In the second part they study strong laws of large numbers for multivariate martingales normalized by linear operators in a finite-dimensional Euclidean space. Corollaries of the general results are considered for martingales under moment restrictions.

Reviewer: Mikhail P. Moklyachuk (Kyïv)

MSC:

60F15	Strong limit theorems
60G42	Martingales with discrete parameter

Keywords:

strong law of large numbers; almost sure convergence; almost sure boundedness; martingales; normalizations by operators