A note on the law of iterated logarithm for weighted sums of random variables
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.
scientific article; zbMATH DE number 3854099