An infinitely divisible distribution involving modified Bessel functions
HTML articles powered by AMS MathViewer
- by Mourad E. H. Ismail and Kenneth S. Miller PDF
- Proc. Amer. Math. Soc. 85 (1982), 233-238 Request permission
Abstract:
We prove that the function \[ {\left ( {\frac {b} {a}} \right )^{\mu - v}}\frac {{{K_\mu }(b{x^{1/2}}){K_v}(a{x^{1/2}})}} {{{K_\mu }(a{x^{1/2}}){K_v}(b{x^{1/2}})}}\] is the Laplace transform of an infinitely divisible probability distribution when $v > \mu \geqslant 0$ and $b > a > 0$. This implies the complete monotonic ity of the function. We also establish a representation as a Stieltjes transform, which implies in particular that the function has positive real part when $x$ lies in the right half-plane. We conjecture that \[ {\left ( {\frac {b} {a}} \right )^{\mu - v}}\frac {{{I_\mu }(a{x^{1/2}}){I_v}(b{x^{1/2}})}} {{{I_\mu }(b{x^{1/2}}){I_v}(a{x^{1/2}})}}\] also is the Laplace transform of an infinitely divisible probability distribution. It is known that in the limit as $v \to \infty$, the infinite divisibility property holds for both functions.References
- Milton Abramowitz and Irene A. Stegun, Handbook of mathematical functions with formulas, graphs, and mathematical tables, National Bureau of Standards Applied Mathematics Series, No. 55, U. S. Government Printing Office, Washington, D.C., 1964. For sale by the Superintendent of Documents. MR 0167642 R. Askey and M. E. H. Ismail, Recurrence relations, continued fractions and orthogonal polynomials (in preparation).
- Richard A. Askey and Mourad E. H. Ismail, The Rogers $q$-ultraspherical polynomials, Approximation theory, III (Proc. Conf., Univ. Texas, Austin, Tex., 1980), Academic Press, New York-London, 1980, pp. 175–182. MR 602713 A. Erdélyi, W. Magnus, F. Oberhettinger and F. G. Tricomi, Higher transcendental functions, vol. 2, McGraw-Hill, New York, 1953.
- William Feller, An introduction to probability theory and its applications. Vol. I, John Wiley & Sons, Inc., New York; Chapman & Hall, Ltd., London, 1957. 2nd ed. MR 0088081
- I. I. Hirschman and D. V. Widder, The convolution transform, Princeton University Press, Princeton, N. J., 1955. MR 0073746
- Mourad E. H. Ismail, Bessel functions and the infinite divisibility of the Student $t$-distribution, Ann. Probability 5 (1977), no. 4, 582–585. MR 448480, DOI 10.1214/aop/1176995766
- Mourad E. H. Ismail, Integral representations and complete monotonicity of various quotients of Bessel functions, Canadian J. Math. 29 (1977), no. 6, 1198–1207. MR 463527, DOI 10.4153/CJM-1977-119-5
- Mourad E. H. Ismail and Douglas H. Kelker, Special functions, Stieltjes transforms and infinite divisibility, SIAM J. Math. Anal. 10 (1979), no. 5, 884–901. MR 541088, DOI 10.1137/0510083
- Mourad E. H. Ismail and C. Ping May, Special functions, infinite divisibility and transcendental equations, Math. Proc. Cambridge Philos. Soc. 85 (1979), no. 3, 453–464. MR 520462, DOI 10.1017/S0305004100055912
- John Kent, Some probabilistic properties of Bessel functions, Ann. Probab. 6 (1978), no. 5, 760–770. MR 0501378
- Kenneth S. Miller, Hypothesis testing with complex distributions, Applied Mathematics Series, Robert E. Krieger Publishing Co., Huntington, N.Y., 1980. MR 564654
- G. N. Watson, A treatise on the theory of Bessel functions, Cambridge Mathematical Library, Cambridge University Press, Cambridge, 1995. Reprint of the second (1944) edition. MR 1349110
- J. G. Wendel, Hitting spheres with Brownian motion, Ann. Probab. 8 (1980), no. 1, 164–169. MR 556423 J. Wimp, Orthogonal polynomials in the tabulation of Stieltjes transforms, J. Math. Anal. Appl. (to appear).
Additional Information
- © Copyright 1982 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 85 (1982), 233-238
- MSC: Primary 60E07; Secondary 33A40
- DOI: https://doi.org/10.1090/S0002-9939-1982-0652449-9
- MathSciNet review: 652449