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August, 1977 Bessel Functions and the Infinite Divisibility of the Student $t$- Distribution
Mourad E. H. Ismail
Ann. Probab. 5(4): 582-585 (August, 1977). DOI: 10.1214/aop/1176995766


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.


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Mourad E. H. Ismail. "Bessel Functions and the Infinite Divisibility of the Student $t$- Distribution." Ann. Probab. 5 (4) 582 - 585, August, 1977.


Published: August, 1977
First available in Project Euclid: 19 April 2007

zbMATH: 0369.60023
MathSciNet: MR448480
Digital Object Identifier: 10.1214/aop/1176995766

Primary: 33A45
Secondary: 60E05

Keywords: Bessel functions , completely monotonic functions , Stieltjes transform , Student $t$-distribution

Rights: Copyright © 1977 Institute of Mathematical Statistics

Vol.5 • No. 4 • August, 1977
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