Theory of Probability and Mathematical Statistics

ISSN 1547-7363(online) ISSN 0094-9000(print)

 

 

The counting process and summation of a random number of random variables


Author: O. V. Sugakova
Translated by: V. Zayats
Original publication: Teoriya Imovirnostei ta Matematichna Statistika, tom 74 (2006).
Journal: Theor. Probability and Math. Statist. 74 (2007), 181-189
MSC (2000): Primary 60F05; Secondary 60K05
Published electronically: July 9, 2007
MathSciNet review: 2336788
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Abstract | References | Similar Articles | Additional Information

Abstract: The behavior of the tail of the sum of a random number of random variables $ \overline{F(x)}=\pr\{\sum_{i=1}^{\nu}\xi_i>x\}$ is considered as $ x \to \infty$. Estimates of the convergence of $ \overline{F(x)}$ to the limit function are constructed in terms of renewal theory. The estimates are based on the variance $ \operatorname{Var}\nu(t)$ of the counting process $ \nu(t)=\min\bigl\{n\colon\sum_{i=1}^n \xi_i>t\bigr\}$. A survey of bounds for $ \operatorname{Var}\nu(t)$ is given for different sequences $ \{\xi_i\}$, in particular, for the case where the terms of the sequence $ \{\xi_i\}$ are not identically distributed.


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Additional Information

O. V. Sugakova
Affiliation: Department of Mathematics and Theoretical Radiophysics, Faculty of Radiophysics, Taras Shevchenko National University, Glushkov Avenue, 2, Building 5, Kyïv 03127, Ukraine
Email: sugak@univ.kiev.ua

DOI: http://dx.doi.org/10.1090/S0094-9000-07-00707-7
Keywords: Nonhomogeneous renewal process, counting process, residual lifetime process, variance of the counting process
Received by editor(s): April 13, 2005
Published electronically: July 9, 2007
Article copyright: © Copyright 2007 American Mathematical Society