On the rate of convergence of series of independent random variables. (English. Russian original) Zbl 0646.41013

Solntsev, S. A.

Theory Probab. Math. Stat. 35, 121-125 (1987); translation from Teor. Veroyatn. Mat. Stat. 35, 105-110 (1986).

Let \(\{X_ n,n\geq 1\}\) be a sequence of independent random variables such that the series \(S:=X_ 1+X_ 2+..\). converges a.s. and let \(a_ n>0\) be a sequence of real numbers with \(a_ n\to \infty\). Consider the partial sums \(S_ n:=X_ 1+...+X_{n-1}\) and \(R_ n:=S-S_ n\). The author gives Borel-Cantelli type conditions which are necessary and sufficient for the relations \(\overline{\lim} a_ nR_ n\leq \alpha\) a.s. and \(=\alpha\) a.s. respectively, as well as for \(a_ nR_ n\to 0\) a.s.

Reviewer: R.Wegmann

MSC:

41A25	Rate of convergence, degree of approximation
60G50	Sums of independent random variables; random walks
60F15	Strong limit theorems

Keywords:

Borel-Cantelli type conditions