On some results of Cufaro Petroni about Student t-processes

TL;DR

This paper investigates the asymptotic behavior and density expansion of Student t-processes, extending previous conjectures and providing explicit results for convolution powers in multivariate cases.

Contribution

It proves and extends conjectures on the asymptotic behavior and density expansion of Student t-processes, including explicit formulas for convolution powers.

Findings

Explicit asymptotic behavior for convolution powers of Student t-densities.

Integer convolution powers with odd degrees are convex combinations of Student t-densities.

Non-integer convolution powers do not share the same convex combination property.

Abstract

This paper deals with Student t-processes as studied in (Cufaro Petroni N 2007 J. Phys. A, Math. Theor. 40(10), 2227-2250). We prove and extend some conjectures expressed by Cufaro Petroni about the asymptotical behavior of a Student t-process and the expansion of its density. First, the explicit asymptotic behavior of any real positive convolution power of a Student t-density with any real positive degrees of freedom is given in the multivariate case; then the integer convolution power of a Student t-distribution with odd degrees of freedom is shown to be a convex combination of Student t-densities with odd degrees of freedom. At last, we show that this result does not extend to the case of non-integer convolution powers.