Estimates of the rate of convergence in the strong law of large numbers (Q1058772)

Consider a sequence of independent random variables \(X_ 1,X_ 2,..\).. \(S_ n=X_ 1+...+X_ n\), \(F_ k(x)=P\{X_ k<x\}\), \(k=1,2,..\). In the present note we investigate the rate of convergence to zero as \(n\to \infty\) of probabilities of the form \[ P\{\sup_{k\geq n}| (S_ k/b_ k)-A_ k| >\epsilon \}, \] where \(\epsilon >0\), \(b_ n>0\) and \(A_ n\) are certain sequences. Analogous investigations for the probabilities \(P\{| s_ n/b_ n-A_ n| >\epsilon \}\) have been carried out by the author, Litov. Mat. Sb. 20, No.4, 147-163 (1980; Zbl 0463.60029). There one can find a bibliography on this topic.