A strong Tauberian theorem for characteristic functions

R Riedi (1) , Paulo Gonçalves (2)

1 UNIFR - Université de Fribourg = University of Fribourg
2 DANTE - Dynamic Networks : Temporal and Structural Capture Approach

Using wavelet analysis we show that if the characteristic function of a random variable X can be approximated at 0 by some polynomial of even degree 2p then the moment of order 2p of X exists. This strengthens a Tauberian-type result by Ramachandran and implies that the characteristic function is actually 2p times differentiable at 0. This fact also provides the theoretical basis for a wavelet based non-parametric estimator of the tail index of a distribution.

Domaines: Traitement du signal et de l'image [eess.SP]

R Riedi, Paulo Gonçalves. A strong Tauberian theorem for characteristic functions. Applied and Computational Harmonic Analysis, 2015, 39, pp.6. ⟨10.1016/j.acha.2015.03.007⟩. ⟨hal-01245436⟩