Recovering distribution functions, interval probabilities and jumps via Fourier inversion theorems: new convergence bounds

Abstract

We provide new Fourier inversion theorems, with rates, that allow the recovery of a distribution function, associated interval probabilities, and jumps from the characteristic function.  The results expand, improve, and clarify conditions imposed in our earlier work Mynbaev et al. (2022). First, we show that higher rates of convergence can be achieved by an appropriate choice of the regularization function, both for the recovery of interval probabilities and for jump discontinuities. Second, we propose a new inversion theorem for the recovery of the distribution function at points of continuity. Along the way, we clarify which of the conditions used previously are in fact necessary. The resulting theorems may be useful for constructing nonparametric estimators in errors-in-variables models where density functions may fail to exist.