The counting process and summation of a random number of random variables
Author:
O. V. Sugakova
Translated by:
V. Zayats
Journal:
Theor. Probability and Math. Statist. 74 (2007), 181-189
MSC (2000):
Primary 60F05; Secondary 60K05
DOI:
https://doi.org/10.1090/S0094-9000-07-00707-7
Published electronically:
July 9, 2007
MathSciNet review:
2336788
Full-text PDF Free Access
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Abstract: The behavior of the tail of the sum of a random number of random variables $\overline {F(x)}=\mathsf P\{\sum _{i=1}^{\nu }\xi _i>x\}$ 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.
References
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References
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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
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