A note on the law of iterated logarithm for weighted sums of random variables. (English) Zbl 0537.60023

Stadtmüller, Ulrich

Ann. Probab. 12, 35-44 (1984).

We give sufficient conditions for the upper and lower inequality in the law of iterated logarithm for weighted sums \(W_ n=\sum_{k}a_{nk}X_ k\) of i.i.d. random variables. The main condition for the upper inequality is: \(Cor(W_ n,W_ m)\to 1\) as n,\(m\to \infty\), s.t. n and m are close to each other; whereas for the lower inequality we demand that the new r.v.s in \(W_{n+\ell}^ w.\)r. to \(W_ n\) should receive enough weight. In addition the variation within the rows \((a_{nk})\) should not be too large. The proof uses explicit calculations for normal r.v.s and strong approximation results for sums of r.v.s.

MSC:

60F15	Strong limit theorems
40C05	Matrix methods for summability

Keywords:

law of iterated logarithm; weighted sums; strong approximation for sums; summability matrix; Borel property