Proceedings of the American Mathematical Society

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ISSN 1088-6826(e) ISSN 0002-9939(p)

Lois du logarithme itéré avec pondérations additives

Author(s): Gérald Tenenbaum
Journal: Proc. Amer. Math. Soc. 137 (2009), 4255-4257.
MSC (2000): Primary 60F15, 11N37; Secondary 60G50
Posted: August 3, 2009
MathSciNet review: 2538586
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Abstract | References | Similar articles | Additional information

Abstract: We provide a very short proof that natural Lindeberg type conditions on the non-negative arithmetic additive function ensure the strong law of large numbers and the law of the iterated logarithm for weighted sums $\sum_{n\leqslant N}f(n)X_n$ for any sequence $\{X_n\}_{n=1}^\infty$ of i.i.d. random variables.

References:

1.: I. Berkes et M. Weber, A law of the iterated logarithm for arithmetic functions, Proc. Amer. Math. Soc. 135, no. 4 (2007), 1223-1232. MR 2262929 (2007j:60043)
2.: E. Fisher, A Skorohod representation and an invariance principle for sums of weighted i.i.d. random variables, Rocky Mountain J. Math. 22, no. 1 (1992), 169-179. MR 1159950 (93e:60065)
3.: K. Fukuyama et Y. Komatsu, A law of large numbers for arithmetic functions, Proc. Amer. Math. Soc. 137, no. 1 (2009), 349-352. MR 2439459
4.: B. Jamison, S. Orey et W. Pruitt, Convergence of weighted averages of independent random variables, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 4 (1965), 40-44. MR 0182044 (31:6268)
5.: G. Tenenbaum, Introduction à la théorie analytique et probabiliste des nombres, troisième édition, coll. Échelles, Paris, 2008.

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Additional Information:

Gérald Tenenbaum
Affiliation: Institut Élie Cartan, Université Henri Poincaré-Nancy 1, BP 239, 54506 Vandouvre lèes Nancy Cedex, France
Email: gerald.tenenbaum@iecn.u-nancy.fr

DOI: 10.1090/S0002-9939-09-10035-7
PII: S 0002-9939(09)10035-7
Received by editor(s): April 25, 2009,
Received by editor(s) in revised form: April 29, 2009
Posted: August 3, 2009
Communicated by: Richard C. Bradley
Copyright of article: Copyright 2009, American Mathematical Society
The copyright for this article reverts to public domain after 28 years from publication.

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