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Strong laws of large numbers for arrays of orthogonal random variables. (English) Zbl 0674.60006

Let H be a separable Hilbert space and let \(\{X_{nk}:\) \(k=1,2,...,n\); \(n=1,2,...,\}\) be a sequence of random variables such that \[ \sigma^ 2_{nk}:=E[\| X_{nk}\|^ 2]<\infty,\quad and\quad E[(X_{nk},X_{nj})]=0,\quad k\neq j. \] Such a sequence is called a row-wise orthogonal array. For fixed \(\alpha >0\) the means \(\zeta_ n\) are defined by \(\zeta_ n=n^{-\alpha}\sum^{n}_{k=1}X_{nk},\) and such a sequence is said to converge to zero completely if, \[ \forall \epsilon >0:\quad \sum^{\infty}_{n=1}P[\| \zeta_ n\| >\epsilon]<\infty. \] The authors show firstly that, if \[ \sum^{\infty}_{n=1}n^{-2\alpha}\sum^{n}_{k=1}\sigma^ 2_{nk}<\infty, \] for some \(\alpha >0\), then \(\zeta_ n\) converges to zero completely. This is shown to be the best possible result even for \(H={\mathbb{R}}\) and for orthogonality between, as well as within, any two rows in \(\{X_{nk}\}.\)
Various extensions (e.g. for generalized arrays, for any sequence (\(\lambda\) (n)) of positive numbers rather than \(n^{\alpha})\) are discussed, and a study of the problems presented by orthogonality in Banach spaces is made.
Reviewer: A.Dale

MSC:

60B12 Limit theorems for vector-valued random variables (infinite-dimensional case)
46C99 Inner product spaces and their generalizations, Hilbert spaces
60F15 Strong limit theorems
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References:

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