## The Annals of Probability

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

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

#### 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

**Permanent link to this document**

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**

links.jstor.org

**Subjects**

Primary: 62E10: Characterization and structure theory

Secondary: 26A48: Monotonic functions, generalizations 33A45 44A10: Laplace transform

**Keywords**

Student $t$-distribution infinite divisibility complete monotonicity Laplace transform Bernstein's theorem

#### Citation

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