Empirical characteristic function
= Empirical characteristic function
{title2=$\varphi_n(u)$}
{wiki}
For a sample $X_1,\ldots,X_n$, $\varphi_n(u)=n^{-1}\sum_j e^{iuX_j}$ is the <characteristic function> of the sample's <empirical measure>. For <independent and identically distributed random variables>, $\mathbb E\varphi_n(u)=\varphi(u)$ and $\operatorname{Var}_{\mathbb C}\varphi_n(u)=(1-|\varphi(u)|^2)/n$, without any moment condition on the observations.