Using the representation theorem and inversion formula for Stieltjes transforms, we give a simple proof of the infinite divisibility of the student $t$-distribution for all degrees of freedom by showing that $x^{-\frac{1}{2}}K_\nu(x^{\frac{1}{2}})/K_{\nu+1}(x^{\frac{1}{2}})$ is completely monotonic for $\nu \geqq -1$. Our approach proves the stronger and new result, that $x^{-\frac{1}{2}}K_\nu (x^{\frac{1}{2}}) /K_{\nu+1}(x^{\frac{1}{2}})$ is a completely monotonic function of $x$ for all real $\nu$. We also derive a new integral representation.
@article{1176995766,
author = {Ismail, Mourad E. H.},
title = {Bessel Functions and the Infinite Divisibility of the Student $t$- Distribution},
journal = {Ann. Probab.},
volume = {5},
number = {6},
year = {1977},
pages = { 582-585},
language = {en},
url = {http://dml.mathdoc.fr/item/1176995766}
}
Ismail, Mourad E. H. Bessel Functions and the Infinite Divisibility of the Student $t$- Distribution. Ann. Probab., Tome 5 (1977) no. 6, pp. 582-585. http://gdmltest.u-ga.fr/item/1176995766/