## Counting process and summation of random number of random variables.(Ukrainian, English)Zbl 1150.60327

Teor. Jmovirn. Mat. Stat. 74, 159-166 (2006); translation in Theory Probab. Math. Stat. 74, 181-189 (2007).
Let $$\{\xi_{i}, i\geq1\}$$ be a sequence of random variables, $$E\xi_{i}>0$$, $$E\xi_{i}^2<\infty$$, $$\forall i\geq1$$. The integer-valued random variable $$\nu$$, $$P\{\nu=n\}=p_{n}$$ is independent on $$\{\xi_{i}\}$$. Let us denote $$\nu(t)=\min\{n:\;\sum_{i=1}^{n}\xi_{i}>t\}$$, $$H(t)=E\nu(t)$$, $$D(t)=D\nu(t)$$, $$\tau=\sum_{i=1}^{\nu}\xi_{i}$$. The author proves the following theorem.
Let $$p(x)$$ be a convex, differentiable, non-increasing function, such that $$p(n)=p_{n}, n\geq0$$. Then for arbitrary $$\alpha\in(0,1)$$ the following estimate holds true $$| P\{\tau>x\}-P\{\nu>H(x)\}|\leq2p(\alpha H(x))+{3+2E\nu\over(1-\alpha)^2}\cdot{D(x)\over H^2(x)}+{1\over2}R(\alpha H(x))(D(x)+1)$$, where $$R(x)=-p'(x)$$.
The bounds of $$D\nu(t)$$ for different sequences $$\{\xi_{i}\}$$ are presented.

### MSC:

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