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Theory of Probability and Mathematical Statistics
Theory of Probability and Mathematical Statistics
ISSN 1547-7363(online) ISSN 0094-9000(print)

 

Improvement of the stability of solutions of an inhomogeneous perturbed renewal equation on the semiaxis


Author: M. V. Kartashov
Translated by: O. I. Klesov
Original publication: Teoriya Imovirnostei ta Matematichna Statistika, tom 84 (2011).
Journal: Theor. Probability and Math. Statist. 84 (2012), 65-78
MSC (2010): Primary 60J45; Secondary 60A05, 60K05
Published electronically: July 31, 2012
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Abstract: We consider a generalized inhomogeneous continuous-time renewal equation on the semiaxis that reduces to the Volterra integral equation with a nonnegative bounded (or substochastic) kernel. It is assumed that the kernel can be approximated in a large time scale by a convolution kernel generated by a stochastic distribution. Under some asymptotic assumptions imposed on the perturbation we find an improved condition for the boundedness; the latter condition is used to prove the existence of the limit of a solution of the perturbed equation and to establish estimates for the deviation from a solution of a nonperturbed equation.

Some examples are discussed.


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Additional Information

M. V. Kartashov
Affiliation: Department of Probability Theory, Statistics, and Actuarial Mathematics, Faculty for Mechanics and Mathematics, National Taras Shevchenko University, Academician Glushkov Avenue 4E, Kiev 03127, Ukraine
Email: nkartashov@skif.com.ua

DOI: http://dx.doi.org/10.1090/S0094-9000-2012-00854-5
PII: S 0094-9000(2012)00854-5
Keywords: Volterra equation, renewal theory, transition kernel, minimal solution, stability
Received by editor(s): September 3, 2010
Published electronically: July 31, 2012
Article copyright: © Copyright 2012 American Mathematical Society