Central moments of a nonhomogeneous renewal process (Q2737035)

Let \(\{\xi_{i}\); \(i\geq 0\}\) be a sequence of i.i.d. random variables. It is supposed that \(\xi_{i}\) have uniformly bounded moments of \(m\)th order, \(m\geq 2.\) This paper deals with the nonhomogeneous renewal process generated by the sequence \(\xi_{i}.\) Let \(\nu(t)\) be a counting process generated by \(\{\xi_{i}\}\) and let \(H(t)=E\nu(t)\) be a renewal function. The estimation for central moments \(E(\nu(t)-H(t))^{m}\) of the counting process \(\nu(t)\) is derived. The obtained results are applied to estimation of the rate of convergence in the problem of summing of a geometric number of random variables.