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

Sugakova, O. V.

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.

Reviewer: Aleksandr D. Borisenko (Kyïv)

MSC:

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

Keywords:

counting process; summation of random number of random variables; nonhomogeneous renewal process