A minimal uniform renewal theorem and transition phenomena for a nonhomogeneous perturbation of the renewal equation
Author:
M. V. Kartashov
Translated by:
S. Kvasko
Journal:
Theor. Probability and Math. Statist. 82 (2011), 27-41
MSC (2010):
Primary 60J45; Secondary 60A05, 60K05
DOI:
https://doi.org/10.1090/S0094-9000-2011-00825-3
Published electronically:
August 2, 2011
MathSciNet review:
2790481
Full-text PDF Free Access
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.
References
- N. V. Kartashov, Strong stable Markov chains, VSP, Utrecht; TBiMC Scientific Publishers, Kiev, 1996. MR 1451375
- M. V. Kartashov, Inhomogeneous perturbations of a renewal equation and the Cramér-Lundberg theorem for a risk process with variable premium rates, Teor. Ĭmovīr. Mat. Stat. 78 (2008), 54–65 (Ukrainian, with Ukrainian summary); English transl., Theory Probab. Math. Statist. 78 (2009), 61–73. MR 2446849, DOI https://doi.org/10.1090/S0094-9000-09-00762-5
- M. V. Kartashov, Boundedness, limits, and stability of solutions of an inhomogeneous perturbation of a renewal equation on a half-line, Teor. Ĭmovīr. Mat. Stat. 81 (2009), 65–75 (Ukrainian, with English and Ukrainian summaries); English transl., Theory Probab. Math. Statist. 81 (2010), 71–83. MR 2667311, DOI https://doi.org/10.1090/S0094-9000-2010-00811-8
- D. C. M. Dickson, The probability of ultimate ruin with a variable premium loading — a special case, Scand. Actuarial J. (1991), 75–86.
- Hans U. Gerber, On the probability of ruin in the presence of a linear dividend barrier, Scand. Actuar. J. 2 (1981), 105–115. MR 623405, DOI https://doi.org/10.1080/03461238.1981.10413735
- H. Gerber and E. S. W. Shiu, On the time value of ruin, Proc. 31 Actuarial Research Conference, Ball Statte Univ., August 1996, pp. 145–199.
- H. Schmidli, An extension to the renewal theorem and an application to risk theory, Ann. Appl. Probab. 7 (1997), no. 1, 121–133. MR 1428752, DOI https://doi.org/10.1214/aoap/1034625255
- G. C. Taylor, Probability of ruin with variable premium rate, Scand. Actuar. J. 2 (1980), 57–76. MR 578447, DOI https://doi.org/10.1080/03461238.1980.10408641
- M. V. Kartashov, On ruin probabilities for a risk process with bounded reserves, Teor. Ĭmovīr. Mat. Stat. 60 (1999), 46–58 (Ukrainian, with Ukrainian summary); English transl., Theory Probab. Math. Statist. 60 (2000), 53–65 (2001). MR 1826141
- M. V. Kartashov and O. M. Stroēv, The Lundberg approximation for the risk function in an almost homogeneous environment, Teor. Ĭmovīr. Mat. Stat. 73 (2005), 63–71 (Ukrainian, with Ukrainian summary); English transl., Theory Probab. Math. Statist. 73 (2006), 71–79. MR 2213842, DOI https://doi.org/10.1090/S0094-9000-07-00682-5
- M. V. Kartashov, Stability of transient quasihomogeneous Markov semigroups and an estimate for ruin probability, Teor. Ĭmovīr. Mat. Stat. 75 (2006), 36–44 (Ukrainian, with Ukrainian summary); English transl., Theory Probab. Math. Statist. 75 (2007), 41–50. MR 2321179, DOI https://doi.org/10.1090/S0094-9000-08-00712-6
- N. V. Kartashov, Equivalence of uniform renewal theorems and their criteria, Teor. Veroyatnost. i Mat. Statist. 27 (1982), 51–60, 158 (Russian). MR 673349
- N. V. Kartashov, A generalization of Stone’s representation and necessary conditions for uniform convergence in the renewal theorem, Teor. Veroyatnost. i Mat. Statist. 26 (1982), 49–62, 158 (Russian). MR 664903
- William Feller, An introduction to probability theory and its applications. Vol. II, John Wiley & Sons, Inc., New York-London-Sydney, 1966. MR 0210154
- A. A. Borovkov, Veroyatnostnye protsessy v teorii massovogo obsluzhivaniya, Izdat. “Nauka”, Moscow, 1972 (Russian). MR 0315800
- Charles Stone, On absolutely continuous components and renewal theory, Ann. Math. Statist. 37 (1966), 271–275. MR 196795, DOI https://doi.org/10.1214/aoms/1177699617
- V. M. Shurenkov, Ergodic theorems and related problems, VSP, Utrecht, 1998. MR 1690361
References
- N. V. Kartashov, Strong Stable Markov Chains, VSP/TViMS Scientific Publishers, Utrecht/Kiev, 1996. MR 1451375 (99e:60150)
- M. V. Kartashov, Inhomogeneous perturbations of a renewal equation and the Cramér–Lundberg theorem for a risk process with variable premium rates, Teor. Imovir. Mat. Stat. 78 (2008), 55–66; English transl. in Theory Probab. Math. Statist. 78 (2009), 61–73. MR 2446849 (2010a:60295)
- M. V. Kartashov, Boundedness, limits, and stability of solutions of renewal equation with a nonhomogeneous perturbation on the semiaxis, Teor. Imovir. Mat. Stat. 81 (2009), 65–75; English transl. in Theory Probab. Math. Statist. 81 (2010), 71–83. MR 2667311 (2011f:60154)
- D. C. M. Dickson, The probability of ultimate ruin with a variable premium loading — a special case, Scand. Actuarial J. (1991), 75–86.
- H. Gerber, On the probability of ruin in the presence of a linear dividend barrier, Scand. Actuarial J. (1981), 105–115. MR 623405 (83c:62169)
- H. Gerber and E. S. W. Shiu, On the time value of ruin, Proc. 31 Actuarial Research Conference, Ball Statte Univ., August 1996, pp. 145–199.
- H. Schmidli, An extension to the renewal theorem and an application to risk theory, Ann. Appl. Probab. 7 (1997), no. 1, 121–133. MR 1428752 (97k:60234)
- G. C. Tailor, Probability of ruin with variable premium rate, Scand. Actuarial J. (1980), 57–76. MR 578447 (81m:62188)
- N. V. Kartashov, On ruin probabilities for a risk process with bounded reserves, Teor. Imovir. Mat. Stat. 60 (1999), 46–58; English transl. in Theor. Probab. Math. Stat. 60 (2000), 53–65. MR 1826141
- M. V. Kartashov and O. M. Stroyev, The Lundberg approximation for the risk function in an almost homogeneous environment, Teor. Imovir. Mat. Stat. 73 (2005), 63–71; English transl. in Theor. Probab. Math. Stat. 73 (2006), 71–79. MR 2213842 (2007b:62121)
- M. V. Kartashov, Stability of transient quasihomogeneous Markov semigroups and an estimate for ruin probability, Teor. Imovir. Mat. Stat. 75 (2006), 36–44; English transl. in Theor. Probab. Math. Stat. 75 (2007), 41–50. MR 2321179 (2008d:60100)
- N. V. Kartashov, Equivalence of uniform renewal theorems and their criteria, Teor. Veroyatnost. i Mat. Statist. 27 (1982), 51–60; English transl. in Theor. Probab. Math. Stat. 27 (1983), 55–64. MR 673349 (83m:60114)
- N. V. Kartashov, A generalization of the Stone representation and necessary conditions for uniform convergence in the renewal theorem, Teor. Veroyatnost. i Mat. Statist. 26 (1982), 49–62; English transl. in Theor. Probab. Math. Stat. 26 (1983), 53–67. MR 664903 (83m:60113)
- W. Feller, An Introduction to Probability Theory and its Applications, vol. 2, John Wiley & Sons, New York, 1966. MR 0210154 (35:1048)
- A. A. Borovkov, Stochastic Processes in Queueing Theory, Nauka, Moscow, 1972; English transl., Springer-Verlag, New York–Berlin, 1976. MR 0315800 (47:4349)
- C. Stone, On absolutely continuous components and renewal theory, Ann. Math. Stat. 37 (1966), 271–275. MR 0196795 (33:4981)
- V. M. Shurenkov, Ergodic Theorems and Related Problems, Nauka, Moscow, 1989; English transl., VSP, Utrecht, 1998. MR 1690361 (2000i:60002)
Similar Articles
Retrieve articles in Theory of Probability and Mathematical Statistics
with MSC (2010):
60J45,
60A05,
60K05
Retrieve articles in all journals
with MSC (2010):
60J45,
60A05,
60K05
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
Email:
nkartashov@skif.com.ua
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