Limit theorems for random vectors with operator normalizations. I. (English. Ukrainian original) Zbl 0986.60021

Koval’, V. O.

Theory Probab. Math. Stat. 61, 49-60 (2000); translation from Teor. Jmovirn. Mat. Stat. 61, 47-58 (2000).

Let \(\{S_{n}, n\geq 0\}\) be a sequence of random vectors in \(R^{m},\) and let \(\{A_{n}, n\geq 1\}\) be a sequence of non-random linear operators from \(R^{m}\) to \(R^{d},\) \(m,d\geq 1.\) This paper deals with the general theorem on the strong law of large numbers (LLN) with operator normalizing for arbitrary sequences of random vectors \(S_{n},\) namely, it is proved that \(\|A_{n}S_{n}\|\to 0\) a.s. as \(n\to\infty,\) under some conditions, where \(\|\cdot\|\) is the Euclidian norm. New results on the strong LLN for sequences of sums of independent and orthogonal random vectors and for vector martingales are derived with the help of this general theorem. The main attention is paid to the vector martingales.

Reviewer: A.V.Swishchuk (Kyïv)

MSC:

60F15	Strong limit theorems
60B12	Limit theorems for vector-valued random variables (infinite-dimensional case)
60G42	Martingales with discrete parameter

Keywords:

operator-normalized random vectors; limit theorems; vector martingales; law of large numbers