For a sample , is the characteristic function of the sample's empirical measure. For independent and identically distributed random variables, and , without any moment condition on the observations.
The centered process describes fluctuations of the empirical characteristic function. At a fixed frequency, its real and imaginary parts satisfy a multivariate central limit theorem. A process-level limit requires further control over the frequency index.
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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.