Limit Behaviour of the Empirical Characteristic Function

The convergence properties of the empirical characteristic process Yₙ (t) = n^1/2 (cₙ (t) - c (t) ) are investigated. The finite-dimensional distributions of Yₙ 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 Yₙ cannot converge weakly. When the underlying distribution has high enough moments then Yₙ 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 Yₙ are derived. A Strassen-type log log law is established for Yₙ, 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.