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A law of large numbers for arithmetic functions


Authors: Katusi Fukuyama and Yutaka Komatsu
Journal: Proc. Amer. Math. Soc. 137 (2009), 349-352
MSC (2000): Primary 60F15, 11A25; Secondary 60G50
DOI: https://doi.org/10.1090/S0002-9939-08-09517-8
Published electronically: August 15, 2008
MathSciNet review: 2439459
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Abstract | References | Similar Articles | 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 [Enhancements On Off] (What's this?)

  • 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: https://doi.org/10.1090/S0002-9939-08-09517-8
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
Published electronically: 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
Article copyright: © Copyright 2008 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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