The Empirical Characteristic Function and Its Applications

Certain probability properties of cₙ(t), the empirical characteristic function (ecf) are investigated. More specifically it is shown under some general restrictions that cₙ(t) converges uniformly almost surely to the population characteristic function $c(t).$ The weak convergence of n1/2(cₙ(t) - c(t)) to a Gaussian complex process is proved. It is suggested that the ecf may be a useful tool in numerous statistical problems. Application of these ideas is illustrated with reference to testing for symmetry about the origin: the statistic ∫ cₙ(t)² dG(t) is proposed and its asymptotic distribution evaluated.