Remote Access Theory of Probability and Mathematical Statistics

Theory of Probability and Mathematical Statistics

ISSN 1547-7363(online) ISSN 0094-9000(print)

 
 

 

The Erdös-Rényi law for renewal processes


Author: A. N. Frolov
Translated by: The author
Original publication: Teoriya Imovirnostei ta Matematichna Statistika, tom 68 (2003).
Journal: Theor. Probability and Math. Statist. 68 (2004), 157-166
MSC (2000): Primary 60F15; Secondary 60K05
DOI: https://doi.org/10.1090/S0094-9000-04-00593-9
Published electronically: May 11, 2004
MathSciNet review: 2000645
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: The Erdös-Rényi law and strong law of large numbers are proved for renewal processes constructed from nonidentically distributed random variables.


References [Enhancements On Off] (What's this?)

  • 1. A. A. Borovkov, Probability theory, ``Nauka'', Moscow, 1986; English transl., Gordon and Breach, Amsterdam, 1998. MR 88c:60001; MR 2000f:60001
  • 2. B. V. Gnedenko, The theory of probability, 4th edition, ``Nauka'', Moscow, 1988; English transl. of 3rd edition, Chelsea, New York, 1984. MR 25:2622
  • 3. W. Feller, An introduction to probability theory and its applications, Vol. 2, Wiley, New York, 1971. MR 42:5292
  • 4. A. Gut, Stopped random walks. Limit theorems and applications, Springer, New York-Berlin-Heidelberg, 1988. MR 88m:60085
  • 5. A. Gut, O. Klesov, and J. Steinebach, Equivalencies in strong limit theorems for renewal counting processes, Statist. Probab. Lett. 35 (1997), 381-394. MR 98m:60043
  • 6. A. N. Frolov, A. I. Martikainen, and J. Steinebach, Limit theorems for maxima of sums and renewal processes, Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov (LOMI) 278 (2001), 261-274; English transl., J. Math. Sci. 118 (2004), no. 16, 5658-5666. MR 2002j:60049
  • 7. P. Erdös and A. Rényi, On a new law of large numbers, J. Analyse Math. 23 (1970), 103-111. MR 42:6907
  • 8. J. Steinebach, Improved Erdös-Rényi and strong approximation laws for increments of renewal processes, Ann. Probab. 14 (1986), 547-559. MR 87k:60090
  • 9. P. Deheuvels and J. Steinebach, Sharp rates for increments of renewal processes, Ann. Probab. 17 (1989), 700-722. MR 90a:60058
  • 10. J.-N. Bacro, P. Deheuvels, and J. Steinebach, Exact convergence rates in Erdös-Rényi type theorems for renewal processes, Ann. Inst. Henri Poincaré 23 (1987), 195-207. MR 88k:60055
  • 11. J. Steinebach, Strong laws for small increments of renewal processes, Ann. Probab. 19 (1991), 1768-1776. MR 92k:60071
  • 12. V. V. Petrov, On the probabilities of large deviations for sums of independent random variables, Teor. Veroyatnost. i Primenen. 10 (1965), no. 2, 310-322; English transl., Theory Probab. Appl. X (1965), no. 2, 287-298. MR 32:3107
  • 13. P. Deheuvels, L. Devroye, and J. Lynch, Exact convergence rate in the limit theorems of Erdös-Rényi and Shepp, Ann. Probab. 14 (1986), 209-223. MR 87d:60032
  • 14. A. N. Frolov, A. I. Martikainen, and J. Steinebach, Erdös-Rényi-Shepp type laws in the non-i.i.d. case, Studia Sci. Math. Hungarica 33 (1997), 127-151. MR 98f:60058

Similar Articles

Retrieve articles in Theory of Probability and Mathematical Statistics with MSC (2000): 60F15, 60K05

Retrieve articles in all journals with MSC (2000): 60F15, 60K05


Additional Information

A. N. Frolov
Affiliation: Department of Mathematics and Mechanics, St. Petersburg State University, Bibliotechnaya Pl. 2, Staryi Petergof, St. Petersburg 198904, Russia
Email: Andrei.Frolov@pobox.spbu.ru

DOI: https://doi.org/10.1090/S0094-9000-04-00593-9
Keywords: Renewal processes, increments, Erd\"os--R\'enyi law of large numbers
Received by editor(s): April 4, 2002
Published electronically: May 11, 2004
Additional Notes: Partially supported by RFFI, grant 02-01-00779, and Ministry of Education of the Russian Federation, grant E00-1.0-82
Article copyright: © Copyright 2004 American Mathematical Society

American Mathematical Society