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

Klesov, O. I.

Teor. Veroyatn. Mat. Stat., Kiev 39, 65-71 (1988).

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

Citations:

Zbl 0536.60036