Stieltjes functions of finite order and hyperbolic monotonicity

Abstract

A class of Stieltjes functions of finite type is introduced. These satisfy Widder's conditions on the successive derivatives up to some finite order, and are not necessarily smooth. We show that such functions have a unique integral representation, along some generic kernel which is a truncated Laurent series approximating the standard Stieltjes kernel. We then obtain a two-to-one correspondence, via the logarithmic derivative, between these functions and a subclass of hyperbolically monotone functions of finite type. This correspondence generalizes a representation of HCM functions in terms of two Stieltjes transforms earlier obtained by the first author.2010 Mathematics Subject Classification. 44A15, 60E10.

Estimation of high dimensional Gamma convolutions through random projections

“…While the univariate Thorin class benefits from a large description of its content and properties, e.g., in [25,36], or in [42,27,7,8], the statistical literature and especially regarding the problem of the estimation of such parametric models is a lot sparser. One potential reason is that the deconvolution of an empirical distribution into several Gamma distributions is a hard inverse problem, and the deconvolution literature already has troubles with removing a small convolutional noise from a strong signal.…”