On the complete convergence for randomly selected sequences (Q910808)

Let \(\psi(x)\) and \(H(x)\) be positive functions with \(H(x) \uparrow\), \(x \psi(x)\uparrow \infty\) as \(x\to \infty\). Assume that there exists \(\delta >0\) such that \(H(x)\geq \delta H(2x)\) for each \(x>0\). It is proved that \[ \sum^{\infty}_{n=1}\psi (n)P(\max_{i\leq n}| S_ i| \geq \epsilon H(n))<\infty,\quad \forall \epsilon >0 \] implies \[ \sum^{\infty}_{n=1}[(n\psi (n)-[n/2]\psi ([n/2]))/n]P(\sup_{i\geq \alpha n}(| S_ i| /H(i))\geq \epsilon)<\infty,\quad \forall \epsilon >0,\quad \alpha >0, \] for an arbitrary sequence \(\{X_ n\}\) of random variables, and vice versa for i.i.d. random variables. As an application, the complete convergence of random partial sums of arbitrary random variables is obtained.