×

A proof of Steutel’s conjecture. (English) Zbl 0796.60017

A proof is given of two long-standing conjectures of mine [Preservation of infinite divisibility under mixing and related topics (1970; Zbl 0226.60013), pp. 28-31]. The first conjecture states the infinite divisibility of mixtures of Gamma distributions with shape parameters not exceeding 2; equivalently: densities of the form \(xf(x)\) where \(f\) is completely monotone are infinitely divisible. By an inequality of Karamata, this conjecture would be proved if the following, more complicated conjecture were true. For \(n\geq 2\) let \(A_ j>0\) \((j=1,2,\ldots,n)\) and \(0<\lambda_ 1<\lambda_ 2<\ldots<\lambda_ n\). Let \(z_ j=\mu_ j\pm i\nu_ j\) \((j=1,2,\ldots,n-1)\) be the \(2n-2\) zeroes of the function \(\sum^ n_{j=1} A_ j/(z-\lambda_ j)^ 2\), and let the \(\mu_ j\) be ordered such that \(\mu_ 1\leq\mu_ 2\leq\ldots\leq\mu_{n-1}\). The conjecture states that \[ \sum^ k_{j=1}\lambda_ j<\sum^ k_{j=1}\mu_ j \qquad (k=1,2,\ldots,n- 1). \tag{1} \] The core of the paper is a proof of (1). The proof, which uses induction on \(p\), is ingenious and complicated; it involves complex- function theory and careful estimation of order terms. Reading the proof line by line is hard enough; I have not succeeded in really understanding it. Though I am happy that the conjecture has been settled, I hope that a simpler proof of (1) will be found. For some time I have thought that an inequality, similar to (1), for the eigenvalues and singular values of a matrix might be useful, but I do not see how. The author is to be complimented for having succeeded to give a proof at all.

MSC:

60E07 Infinitely divisible distributions; stable distributions
30C15 Zeros of polynomials, rational functions, and other analytic functions of one complex variable (e.g., zeros of functions with bounded Dirichlet integral)
26C10 Real polynomials: location of zeros
26D99 Inequalities in real analysis

Citations:

Zbl 0226.60013
PDFBibTeX XMLCite
Full Text: DOI