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

Author:
Rita Giuliano

Journal:
Theor. Probability and Math. Statist. **83** (2011), 59-69

MSC (2010):
Primary 60F05, 60G52, 60G70, 11K06; Secondary 62G07, 42A10, 42A61

DOI:
https://doi.org/10.1090/S0094-9000-2012-00841-7

Published electronically:
February 2, 2012

MathSciNet review:
2768848

Full-text PDF Free Access

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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

Email:
giuliano@dm.unipi.it

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