A data-driven Fourier-mixture neural-network method for density estimation

We propose a data-driven Fourier-trained neural-network method for estimating fixed-horizon probability densities from empirical characteristic-function (CF) information. The estimator is a positive Gaussian--Laplace mixture with closed-form CF, so training can be performed directly in Fourier space while preserving nonnegativity and unit mass. We consider two sampling settings. In the direct i.i.d. sampling setting, the method is trained against an empirical CF constructed from i.i.d. samples. In the resampling-based pseudo-sampling setting, it is trained against an empirical pseudo-CF constructed from dependent data by resampling. For the direct i.i.d. case, we derive an expected squared L_2 density-error bound that separates Fourier truncation, empirical training error, discretization, and CF sampling error, together with a corresponding bound for the expected L_2 norm. For the pseudo-sampling case, we obtain conditional counterparts with an additional pseudo-law discrepancy term. We develop a multidimensional extension of the framework and analyze its computational complexity. Numerical experiments show competitive performance relative to Expectation--Maximization on Gaussian-mixture benchmarks and clear gains on heavy-tailed targets. A controlled comparison with exact-CF training quantifies the additional density error introduced by empirical-CF training; further experiments show that the L_2 norm decays approximately as the inverse square root of the sample size when the remaining error terms are controlled, and demonstrate the method on a one-year Australian equity return law estimated from resampled dependent data.