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

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.

### MSC:

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