## The Annals of Probability

### The Student $t$-Distribution for Odd Degrees of Freedom is Infinitely Divisible

Emil Grosswald

#### Abstract

Let $P_n$ be the $n$th Bessel polynomial. Kelker (1971) showed that the Student $t$-distribution of $k = 2n + 1$ degrees of freedom is infinitely divisible if and only if $\varphi_n(x) = P_{n-1}(x^{\frac{1}{2}})/P_n(x^{\frac{1}{2}})$ is completely monotonic. Kelker and Ismail proved that $\varphi_n$ is indeed completely monotonic for some small values of $n$ and conjectured that this is always the case. This conjecture is proved here by a twofold application of Bernstein's theorem and the use of some special properties of the zeros of the Bessel polynomials. The same conclusion follows for $Y_k = (\chi k^2)^{-1}$, where $\chi k^2$ is a chi-square variable with $k$ degrees of freedom.

#### Article information

Source
Ann. Probab., Volume 4, Number 4 (1976), 680-683.

Dates
First available in Project Euclid: 19 April 2007

https://projecteuclid.org/euclid.aop/1176996038

Digital Object Identifier
doi:10.1214/aop/1176996038

Mathematical Reviews number (MathSciNet)
MR410848

Zentralblatt MATH identifier
0339.60008

JSTOR
Grosswald, Emil. The Student $t$-Distribution for Odd Degrees of Freedom is Infinitely Divisible. Ann. Probab. 4 (1976), no. 4, 680--683. doi:10.1214/aop/1176996038. https://projecteuclid.org/euclid.aop/1176996038