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Bernoulli 13(4), 2007, 10911123 DOI: 10.3150/07-BEJ6143
 

Summary: Bernoulli 13(4), 2007, 1091­1123
DOI: 10.3150/07-BEJ6143
Bounds for the covariance of functions of
infinite variance stable random variables
with applications to central limit theorems
and wavelet-based estimation
VLADAS PIPIRAS1, MURAD S. TAQQU2 and PATRICE ABRY3
1Department of Statistics and Operations Research, UNC-CH, CB #3260, Chapel Hill, NC 27599-3260,
USA. E-mail: pipiras@email.unc.edu
2CNRS UMR 5672, Ecole Normale Supérieure de Lyon, Laboratoire de Physique, 69 364 Lyon Cedex 07,
France. E-mail: pabry@physique.ens-lyon.fr
3Department of Mathematics and Statistics, Boston University, 111 Cummington St., Boston, MA 02215,
USA. E-mail: murad@math.bu.edu
We establish bounds for the covariance of a large class of functions of infinite variance stable random vari-
ables, including unbounded functions such as the power function and the logarithm. These bounds involve
measures of dependence between the stable variables, some of which are new. The bounds are also used to
deduce the central limit theorem for unbounded functions of stable moving average time series. This result
extends the earlier results of Tailen Hsing and the authors on central limit theorems for bounded functions
of stable moving averages. It can be used to show asymptotic normality of wavelet-based estimators of the
self-similarity parameter in fractional stable motions.

  

Source: Abry, Patrice - Laboratoire de Physique, Ecole Normale Supérieure de Lyon

 

Collections: Engineering