## Convergence rate for sums of a random number of random variables.(Russian)Zbl 0667.60037

Complete convergence and convergence rates results for sums of random variables have been generalized to the case of randomly indexed partial sums. The following result is obtained.
Let $$X_ n$$ be i.i.d. random variables and $$\nu_ n$$ be nonnegative, integer-valued random variables. Suppose that $$\alpha >1/2$$, $$\alpha$$ $$r\geq 1$$, E $$| X_ 1|^ t<\infty$$ and that E $$X_ 1=0$$ if $$r\geq 1$$. If, for some $$\epsilon >0$$, $$\beta >0$$, $(1)\quad \sum n^{\alpha r-2}P(\nu_ n<n^{\beta})<\infty \quad and\quad (2)\quad \sum n^{\alpha r-2}P(\max_{k\leq \nu_ n}| X_ k| \geq \epsilon \nu_ n^{\alpha})<\infty,$ then there exists a number N such that for $$\delta\geq N\epsilon$$; $\sum n^{\alpha r-2}P(| S_ n| \geq \delta \nu_ n^{\alpha})<\infty.$ This result generalizes the theorem of A. Gut in Acta Math. Hung. 42, 225-232 (1983; Zbl 0536.60036). Note that this result holds if E $$| X_ 1|^ t<\infty$$ for $$t>1/\alpha$$ and E $$X_ 1=0$$ if $$\alpha\leq 1$$, and conditions (1) and (2) are satisfied.
Reviewer: O.I.Klesov

### MSC:

 60F15 Strong limit theorems 60G50 Sums of independent random variables; random walks

### Keywords:

convergence rates; randomly indexed partial sums

Zbl 0536.60036