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Theory of Probability and Mathematical Statistics

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A minimal uniform renewal theorem and transition phenomena for a nonhomogeneous perturbation of the renewal equation

Author: M. V. Kartashov
Translated by: S. Kvasko
Original publication: Teoriya Imovirnostei ta Matematichna Statistika, tom 82 (2010).
Journal: Theor. Probability and Math. Statist. 82 (2011), 27-41
MSC (2010): Primary 60J45; Secondary 60A05, 60K05
Published electronically: August 2, 2011
MathSciNet review: 2790481
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Abstract | References | Similar Articles | Additional Information

Abstract: We consider the classical continuous time nonhomogeneous renewal equation on the half-line, that is, the integral Volterra equation with a nonnegative bounded kernel. It is assumed that the kernel can be approximated by a convolution kernel generated by a strong substochastic distribution in the large time scale. We study the scheme of series (transition phenomena) where the corresponding defect tends to zero and the time scale tends to infinity. We find the limit of a solution of the renewal equation under certain assumptions. This result is based on a minimal renewal theorem in the scheme of series.

Some applications are also considered.

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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: Renewal theory, transition limit theorems, minimal solution, stability, Volterra equations, renewal theorem, regularity, minimal solution, stability
Received by editor(s): January 11, 2010
Published electronically: August 2, 2011
Article copyright: © Copyright 2011 American Mathematical Society