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Theory of Probability and Mathematical Statistics

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Weak convergence of sequences from fractional parts of random variables and applications

Author: Rita Giuliano
Original publication: Teoriya Imovirnostei ta Matematichna Statistika, tom 83 (2010).
Journal: Theor. Probability and Math. Statist. 83 (2011), 59-69
MSC (2010): Primary 60F05, 60G52, 60G70, 11K06; Secondary 62G07, 42A10, 42A61
Published electronically: February 2, 2012
MathSciNet review: 2768848
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Abstract | References | Similar Articles | Additional Information

Abstract: We prove results concerning the weak convergence to the uniform distribution on $ [0,1]$ of sequences $ (Z_n)_{n \geq 1}$ of the form $ Z_n = Y_n \pmod 1= \{Y_n \}$, where $ (Y_n)_{n \geq 1}$ is a general sequence of real random variables. Applications are given: (i) to the case of partial sums of (i.i.d.) random variables having a distribution belonging to the domain of attraction of a stable law; (ii) to the case of sample maxima of i.i.d. random variables.

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Additional Information

Rita Giuliano
Affiliation: Dipartimento di Matematica, \lq\lq L. Tonelli\rq\rq, Largo B. Pontecorvo 5, Pisa 56100, Italy

Keywords: Weak convergence, Weyl criterion, Fourier coefficient, characteristic function, partial sum, sample maximum, uniform distribution, Central Limit Theorem, domain of attraction, stable density, stable law, unimodal density, Benford’s law
Received by editor(s): February 25, 2010
Published electronically: February 2, 2012
Additional Notes: Work partially supported by MURST, Italy. The author wishes to thank G. Grekos and E. Janvresse for some helpful discussions, from which the present investigation has arisen
Article copyright: © Copyright 2012 American Mathematical Society

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