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., Visegrád/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


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