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


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.


Download Citation

Sandor Csorgo. "Limit Behaviour of the Empirical Characteristic Function." Ann. Probab. 9 (1) 130 - 144, February, 1981.


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

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

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

Keywords: continuity of a Gaussian process , Convergence rates , Empirical characteristic process , Fernique , Fernique inequality , Fernique-Marcus-Shepp theorem , Jain and Marcus , Komlos-Major-Tusnady theorem , stochastic integral , Strassen-type log log law , strong approximation , theorems of Dudley , weak convergence

Rights: Copyright © 1981 Institute of Mathematical Statistics

Vol.9 • No. 1 • February, 1981
Back to Top