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On the strong law of large numbers and additive functions

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Abstract

Let f(n) be a strongly additive complex-valued arithmetic function. Under mild conditions on f, we prove the following weighted strong law of large numbers: if X,X 1,X 2, … is any sequence of integrable i.i.d. random variables, then

$$ \mathop {\lim }\limits_{N \to \infty } \frac{{\sum\nolimits_{n = 1}^N {f(n)X_n } }} {{\sum\nolimits_{n = 1}^N {f(n)} }} = \mathbb{E}Xa.s. $$

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Authors and Affiliations

  1. Institute of Statistics, Graz University of Technology, Münzgrabenstrasse 11, A-8010, Graz, Austria

    István Berkes & Wolfgang Müller

  2. IRMA, Université Louis-Pasteur et C.N.R.S., 7 rue René Descartes, 67084, Strasbourg Cedex, France

    Michel Weber

Authors
  1. István Berkes
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  2. Wolfgang Müller
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  3. Michel Weber
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Corresponding author

Correspondence to István Berkes.

Additional information

Dedicated to Endre Csáki and Pál Révész on the occasion of their 75th birthdays

Research supported by FWF grant S9603-N23 and OTKA grants K 67961 and K 81928.

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Berkes, I., Müller, W. & Weber, M. On the strong law of large numbers and additive functions. Period Math Hung 62, 1–12 (2011). https://doi.org/10.1007/s10998-011-5001-7

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  • DOI: https://doi.org/10.1007/s10998-011-5001-7

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