On an infinitely divisible distribution involving modified Bessel functions
Abstract
Mourad E.H. Ismail and Kenneth S. Miller in 1982 conjectured that for ν>μ≥0 and b>a>0 the function x↦ G(x;μ,ν)=(b/a)μ−νIμ(a√x)Iν(b√x)/Iμ(b√x)Iν(a√x), where Iν stands for the modified Bessel function of the first kind, is the Laplace transform of an infinitely divisible probability distribution. Their conjecture is equivalent to the monotonicity property of some exponential sums over the zeros jν,n with respect to the order, where jν,n denotes the nth positive zero of the Bessel function of the first kind of order ν. In this paper our aim is to reduce the problem further to the monotonicity, with respect to ν, of Qν(t)=∑n≥1jν,n2e−t jν,n2, t>0, and to prove this monotonicity. In this way we show that indeed the above function x↦ G(x;μ,ν) is the Laplace transform of an infinitely divisible probability distribution, and the above conjecture is true. The key ingredient in our proof is an integral representation for the derivative of a quotient of modified Bessel functions of the first kind with respect to the order, combined with Bernstein's theorem and a probability tail argument.
Related
- Directional Optimal Sub-Gamma Scales for Infinitely Divisible Laws
- Linear Preservers of Infinitely Divisible Matrices
- Risk diversification for infinitely divisible distributions
- Optimal Sub-Gamma Scales for Spectrally Positive Infinitely Divisible Laws
- Minimal Radial Sub-Gamma Envelopes for Infinitely Divisible Random Vectors
BibTeX
@misc{baricz2026infinitely,
title = {{On an infinitely divisible distribution involving modified Bessel functions}},
author = {Árpád Baricz and Dhivya Prabhu K},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.09779},
eprint = {2610.09779},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.09779},
}
RIS
TY - UNPB
TI - On an infinitely divisible distribution involving modified Bessel functions
AU - Árpád Baricz
AU - Dhivya Prabhu K
PY - 2026
DA - 2026/10/07
PB - arXiv
DO - 10.48550/arxiv.2610.09779
UR - https://arxiv.org/abs/2610.09779
AB - Mourad E.H. Ismail and Kenneth S. Miller in 1982 conjectured that for ν>μ≥0 and b>a>0 the function x↦ G(x;μ,ν)=(b/a)^μ−νI_μ(a√x)I_ν(b√x)/I_μ(b√x)I_ν(a√x), where I_ν stands for the modified Bessel function of the first kind, is the Laplace transform of an infinitely divisible probability distribution. Their conjecture is equivalent to the monotonicity property of some exponential sums over the zeros j_ν,n with respect to the order, where j_ν,n denotes the nth positive zero of the Bessel function of the first kind of order ν. In this paper our aim is to reduce the problem further to the monotonicity, with respect to ν, of Q_ν(t)=∑_n≥1j_ν,n^2e^−t j_ν,n^2, t>0, and to prove this monotonicity. In this way we show that indeed the above function x↦ G(x;μ,ν) is the Laplace transform of an infinitely divisible probability distribution, and the above conjecture is true. The key ingredient in our proof is an integral representation for the derivative of a quotient of modified Bessel functions of the first kind with respect to the order, combined with Bernstein's theorem and a probability tail argument.
ER -