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

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.

**1.**N. V. Kartashov,*Strong stable Markov chains*, VSP, Utrecht; TBiMC Scientific Publishers, Kiev, 1996. MR**1451375****2.**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**, https://doi.org/10.1090/S0094-9000-09-00762-5**3.**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**, https://doi.org/10.1090/S0094-9000-2010-00811-8**4.**D. C. M. Dickson,*The probability of ultimate ruin with a variable premium loading -- a special case*, Scand. Actuarial J. (1991), 75-86.**5.**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**, https://doi.org/10.1080/03461238.1981.10413735**6.**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.**7.**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**, https://doi.org/10.1214/aoap/1034625255**8.**G. C. Taylor,*Probability of ruin with variable premium rate*, Scand. Actuar. J.**2**(1980), 57–76. MR**578447**, https://doi.org/10.1080/03461238.1980.10408641**9.**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****10.**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**, https://doi.org/10.1090/S0094-9000-07-00682-5**11.**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**, https://doi.org/10.1090/S0094-9000-08-00712-6**12.**N. V. Kartashov,*Equivalence of uniform renewal theorems and their criteria*, Teor. Veroyatnost. i Mat. Statist.**27**(1982), 51–60, 158 (Russian). MR**673349****13.**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****14.**William Feller,*An introduction to probability theory and its applications. Vol. II*, John Wiley & Sons, Inc., New York-London-Sydney, 1966. MR**0210154****15.**A. A. Borovkov,*\cyr Veroyatnostnye protsessy v teorii massovogo obsluzhivaniya.*, Izdat. “Nauka”, Moscow, 1972 (Russian). MR**0315800****16.**Charles Stone,*On absolutely continuous components and renewal theory*, Ann. Math. Statist.**37**(1966), 271–275. MR**0196795**, https://doi.org/10.1214/aoms/1177699617**17.**V. M. Shurenkov,*Ergodic theorems and related problems*, VSP, Utrecht, 1998. MR**1690361**

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

DOI:
https://doi.org/10.1090/S0094-9000-2011-00825-3

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