 Empirical characteristic function

For a sample $X_1,\dots ,X_n$, ${\phi }_n(u)=n^{-1}{\sum }_j{e}^{iu{X}_j}$ is the characteristic function of the sample's empirical measure. For independent and identically distributed random variables, $\mathbb{E}{\phi }_n(u)=\phi (u)$ and ${\mathrm{Var}}_{\mathbb{C}}{\phi }_n(u)=(1-|\phi (u){|}^2)/n$, without any moment condition on the observations.

The empirical characteristic function (ECF) is a statistical tool used in the analysis of random variables and processes. It is a nonparametric estimator of the characteristic function of a distribution based on a sample of observations. The characteristic function itself is a complex-valued function that provides useful information about a probability distribution, such as the moments and the behavior of sums of random variables.