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Theory of Probability and Mathematical Statistics

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Boundedness, limits, and stability of solutions of a perturbation of a nonhomogeneous renewal equation on a semiaxis

Author: M. V. Kartashov
Translated by: N. Semenov
Original publication: Teoriya Imovirnostei ta Matematichna Statistika, tom 81 (2010).
Journal: Theor. Probability and Math. Statist. 81 (2010), 71-83
MSC (2010): Primary 60J45; Secondary 60A05, 60K05
Published electronically: January 18, 2011
MathSciNet review: 2667311
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Abstract: We consider a time-nonhomogeneous perturbation of the classical renewal equation with continuous time on the semiaxis that can be reduced to the Volterra integral equation with a nonnegative bounded (or with a substochastic) kernel. We assume that this kernel is approximated by a convolution kernel for large time intervals and that the latter is generated by a substochastic distribution. We find necessary and sufficient conditions for the existence of the limit of a solution of the perturbed equation under the assumption that the corresponding perturbation is small in a certain sense. We also obtain estimates for the deviation of the solutions of the perturbed equations from those of the initial equations.

Several applications are described.

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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 2, Kiev 03127, Ukraine

Keywords: Volterra equation, renewal theorem, transition kernel, regularity, minimal solution, stability
Received by editor(s): October 12, 2009
Published electronically: January 18, 2011
Article copyright: © Copyright 2010 American Mathematical Society

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