## Marcinkiewicz strong law of large numbers for B-valued random variables with multidimensional indices.(English)Zbl 0549.60010

Statistics and probability, Proc. 3rd Pannonian Symp., VisegrĂˇd/Hung. 1982, 53-61 (1984).
[For the entire collection see Zbl 0527.00024.]
Let $$\{X_ n$$, $$n\in {\mathbb{N}}^ d\}$$ be a sequence of independent identically distributed random variables with values in a Banach space B and let $$S_ n=\sum_{k\leq n}X_ k$$ for each n in $${\mathbb{N}}^ d$$, the positive d-dimensional lattice points, $$k\leq n$$ being defined coordinatewise. Write $$| n| =\prod^{n}_{i=1}n_ i$$ if $$n=(n_ 1,...,n_ d)$$. The author proves that for $$1\leq p<2$$, if $$E\{\| X_ 1\|^ p(Log^+\| X_ 1\|)^{d-1}\}<\infty$$, $$S_ n/| n|^ p\to 0$$ almost surely iff $$S_ n/| n|^ p\to 0$$ in probability. In particular, if B is of type p, $$S_ n/| n|^ p\to 0$$ almost surely iff $$E\{X_ 1\}=0$$ and $$E\{\| X_ 1\|^ p(Log^+\| X_ 1\|)^{d-1}\}<\infty$$. These theorems and their proofs extend and combine the d-dimensional indices results on the line and the Banach space result when $$d=1$$.
Reviewer: M.Ledoux

### MSC:

 60B12 Limit theorems for vector-valued random variables (infinite-dimensional case) 60G50 Sums of independent random variables; random walks 60F15 Strong limit theorems 60B11 Probability theory on linear topological spaces

Zbl 0527.00024