Rényi’s theorem and a nonhomogeneous renewal process. (English. Ukrainian original) Zbl 0955.60024

Sugakova, O. V.

Theory Probab. Math. Stat. 60, 181-186 (2000); translation from Teor. Jmovirn. Mat. Stat. 60, 162-166 (1999).

The author considers a sequence of independent nonnegative random variables \(\{ \xi_{i}, i\geq 0\}\) such that \(E\xi_{i}=1\), \(i\geq 0\), and \(\max_{i\geq 0}E\xi_{i}^2 =M<\infty.\) Let \(\nu\) be a random variable with geometrical distribution \(P\{ i=k\}=\theta(1-\theta)^{k}\), \(0<\theta<1\), such that \(\nu\) does not depend on \(\xi_{i}.\) Let \(\theta\tau=\theta\sum_{i=0}^{\nu} \xi_{i}\) and let \(U\) be a random variable such that \(U>0\) and \(P\{ U<x\} =1-e^{-x}\), \(x\geq 0.\) The main result of this paper is the following theorem: Let \(0<\theta<\min(1,2/M).\) Then the following inequality holds true \[ \int\limits_0^{\infty}|P\{ \theta\tau<t\}- P\{ U<t\}|dt\leq 2\theta +{6M\theta^2 \over (1-\theta)(2-M\theta)}. \] This theorem is a generalization of similar theorems for identically distributed random variables.

Reviewer: Yu.V.Kozachenko (Kyïv)

MSC:

60F05	Central limit and other weak theorems
60K05	Renewal theory

Keywords:

renewal process; uniform metric; distribution function; exponential distribution; geometrical distribution