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August, 1976 The Student $t$-Distribution for Odd Degrees of Freedom is Infinitely Divisible
Emil Grosswald
Ann. Probab. 4(4): 680-683 (August, 1976). DOI: 10.1214/aop/1176996038


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.


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Emil Grosswald. "The Student $t$-Distribution for Odd Degrees of Freedom is Infinitely Divisible." Ann. Probab. 4 (4) 680 - 683, August, 1976.


Published: August, 1976
First available in Project Euclid: 19 April 2007

zbMATH: 0339.60008
MathSciNet: MR410848
Digital Object Identifier: 10.1214/aop/1176996038

Primary: 62E10
Secondary: 26A48 , 33A45 , 44A10

Keywords: Bernstein's theorem , complete monotonicity , Infinite divisibility , Laplace transform , Student $t$-distribution

Rights: Copyright © 1976 Institute of Mathematical Statistics

Vol.4 • No. 4 • August, 1976
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