Limit Behaviour of the Empirical Characteristic Function

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.