The Erdös–Rényi law for renewal processes
Author:
A. N. Frolov
Translated by:
The author
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 Free Access
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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
- A. A. Borovkov, Probability theory, “Nauka”, Moscow, 1986; English transl., Gordon and Breach, Amsterdam, 1998. MR 88c:60001; MR 2000f:60001
- B. V. Gnedenko, The theory of probability, 4th edition, “Nauka”, Moscow, 1988; English transl. of 3rd edition, Chelsea, New York, 1984. MR 25:2622
- W. Feller, An introduction to probability theory and its applications, Vol. 2, Wiley, New York, 1971. MR 42:5292
- A. Gut, Stopped random walks. Limit theorems and applications, Springer, New York–Berlin–Heidelberg, 1988. MR 88m:60085
- 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
- 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
- P. Erdös and A. Rényi, On a new law of large numbers, J. Analyse Math. 23 (1970), 103–111. MR 42:6907
- J. Steinebach, Improved Erdös–Rényi and strong approximation laws for increments of renewal processes, Ann. Probab. 14 (1986), 547–559. MR 87k:60090
- P. Deheuvels and J. Steinebach, Sharp rates for increments of renewal processes, Ann. Probab. 17 (1989), 700–722. MR 90a:60058
- 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
- J. Steinebach, Strong laws for small increments of renewal processes, Ann. Probab. 19 (1991), 1768–1776. MR 92k:60071
- 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
- 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
- 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
References
- A. A. Borovkov, Probability theory, “Nauka”, Moscow, 1986; English transl., Gordon and Breach, Amsterdam, 1998. MR 88c:60001; MR 2000f:60001
- B. V. Gnedenko, The theory of probability, 4th edition, “Nauka”, Moscow, 1988; English transl. of 3rd edition, Chelsea, New York, 1984. MR 25:2622
- W. Feller, An introduction to probability theory and its applications, Vol. 2, Wiley, New York, 1971. MR 42:5292
- A. Gut, Stopped random walks. Limit theorems and applications, Springer, New York–Berlin–Heidelberg, 1988. MR 88m:60085
- 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
- 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
- P. Erdös and A. Rényi, On a new law of large numbers, J. Analyse Math. 23 (1970), 103–111. MR 42:6907
- J. Steinebach, Improved Erdös–Rényi and strong approximation laws for increments of renewal processes, Ann. Probab. 14 (1986), 547–559. MR 87k:60090
- P. Deheuvels and J. Steinebach, Sharp rates for increments of renewal processes, Ann. Probab. 17 (1989), 700–722. MR 90a:60058
- 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
- J. Steinebach, Strong laws for small increments of renewal processes, Ann. Probab. 19 (1991), 1768–1776. MR 92k:60071
- 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
- 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
- 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
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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
Keywords:
Renewal processes,
increments,
Erdös–Rényi 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
