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February, 1981 Limit Behaviour of the Empirical Characteristic Function
Sandor Csorgo
Ann. Probab. 9(1): 130-144 (February, 1981). DOI: 10.1214/aop/1176994513

Abstract

The convergence properties of the empirical characteristic process $Y_n(t) = n^{1/2}(c_n(t) - c(t))$ are investigated. The finite-dimensional distributions of $Y_n$ converge to those of a complex Gaussian process $Y$. First the continuity properties of $Y$ are discussed. A class of counterexamples is presented, showing that if the underlying distribution has low logarithmic moments then $Y$ is almost surely discontinuous, and hence $Y_n$ cannot converge weakly. When the underlying distribution has high enough moments then $Y_n$ is strongly approximated by suitable sequences of Gaussian processes with specified rate-functions. The approximation is based on that of Komlos, Major and Tusnady for the empirical process. Convergence speeds for the distribution of functionals of $Y_n$ are derived. A Strassen-type log log law is established for $Y_n$, and supremum-functionals on the appropriate set of limit points are explicitly computed. The technique throughout uses results from the theory of the sample function behaviour of Gaussian processes.

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Sandor Csorgo. "Limit Behaviour of the Empirical Characteristic Function." Ann. Probab. 9 (1) 130 - 144, February, 1981. https://doi.org/10.1214/aop/1176994513

Information

Published: February, 1981
First available in Project Euclid: 19 April 2007

zbMATH: 0453.60025
MathSciNet: MR606802
Digital Object Identifier: 10.1214/aop/1176994513

Subjects:
Primary: 60F05
Secondary: 60E05, 60F15, 60G17, 62G99

Rights: Copyright © 1981 Institute of Mathematical Statistics

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Vol.9 • No. 1 • February, 1981
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