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A law of large numbers for arithmetic functions
Author(s):
Katusi
Fukuyama;
Yutaka
Komatsu
Journal:
Proc. Amer. Math. Soc.
137
(2009),
349-352.
MSC (2000):
Primary 60F15, 11A25;
Secondary 60G50
Posted:
August 15, 2008
MathSciNet review:
2439459
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Additional information
Abstract:
We prove the weighted strong law of large numbers for every integrable i.i.d. sequence where the weights are given by a positive strongly additive function satisfying the Lindeberg condition. This result solves one of the open problems raised in the paper by Berkes and Weber (2007).
References:
-
- 1.
- Berkes, I. and Weber, M., A law of the iterated logarithm for arithmetic functions, Proc. Amer. Math. Soc. 135 (2007) 1223-1232. MR 2262929 (2007j:60043)
- 2.
- Erdős, P. and Kac, M. The Gaussian law of errors in the theory of additive number theoretic functions, Amer. J. Math. 62 (1940) 738-742. MR 0002374 (2:42c)
- 3.
- Jamison, B., Orey, S., and Pruitt, W., Convergence of weighted averages of independent random variables, Z. Wahrsch. verw. Geb. 4 (1965) 40-44 MR 0182044 (31:6268)
- 4.
- Kubilius, J., Probabilistic methods in the theory of numbers, Uspekhi Mat. Nauk (N.S.) 11 (1956), 2(68) 31-66; Amer. Math. Soc. Translations 19 (1962) 47-85. MR 0079025 (18:17d)
- 5.
- Shapiro, H. N., Distribution functions of additive arithmetic functions, Proc. Nat. Acad. Sci. USA 42 (1956) 426-430. MR 0079609 (18:113c)
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Additional Information:
Katusi
Fukuyama
Affiliation:
Department of Mathematics, Kobe University, Rokko, Kobe, 657-8501 Japan
Email:
fukuyama@math.kobe-u.ac.jp
Yutaka
Komatsu
Affiliation:
Graduate School of Science and Technology, Kobe University, Rokko, Kobe, 657-8501 Japan
DOI:
10.1090/S0002-9939-08-09517-8
PII:
S 0002-9939(08)09517-8
Keywords:
Strong law of large numbers,
strongly additive functions
Received by editor(s):
November 5, 2007,
Received by editor(s) in revised form:
December 17, 2007, and January 13, 2008
Posted:
August 15, 2008
Additional Notes:
The first author was supported in part by Grant-in-Aid for Scientific Research (B) 17340029 from the Japan Society for the Promotion of Sciences.
Communicated by:
Richard C. Bradley
Copyright of article:
Copyright
2008,
American Mathematical Society
The copyright for this article reverts to public domain after 28 years from publication.
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