Almost certain convergence in double arrays. (English) Zbl 0548.60028

For double arrays of constants \(\{a_{ni}\), \(1\leq i\leq k_ n,n\geq 1\}\) and i.i.d. random variables \(\{X,X_ i\), \(i\geq 1\}\), conditions are given under which the row sums \(\sum^{k_ n}_{i=1}a_{ni}X_ i\to^{a.c.}0\). Both cases \(k_ n\uparrow\infty \) and \(k_ n=\infty\) are treated. In general, the hypotheses involve a trade-off between moment requirements on X and the magnitude of the \(\{a_{ni}\}\). A Marcinkiewicz-Zygmund type strong law is obtained for the special case \(a_{ni}=a_ i/(\sum^{n}_{j=1}a^ p_ j)^{1/p}\), \(a_ i>0\), \(\sum^{n}_{1}a^ p_ j\to\infty \), \(0<p<2\).


60F15 Strong limit theorems
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