Error Bounds for Numerical Inversion of a Probability Characteristic Function

Given the characteristic function of a continuous real random variable, it is possible to construct a finite Fourier series that approximates the density over any specified finite interval. This series converges in mean square if the density decays faster than |x|-1 and the characteristic function decays faster than |t|-1/2; it converges in minimax if the density decays faster than |x|-1 and the characteristic function decays faster than |t|-1. It is also possible to construct a finite Fourier series for the distribution function that has arbitrarily small maximum error over any finite interval provided that the characteristic function decays as $|t|^{-\beta}$ for some $\beta>0$ and that either the distribution or the density decays (to 0 or 1, as appropriate) as $|x|^{-\alpha}$ for some $\alpha>1$; these conditions are weak enough to include most continuous distributions of practical interest. Explicit error bounds can be obtained directly from the characteristic function, without prior knowledge of either the distribution or density, making these series useful for numerical inversion.