A Note on Infinite Divisibility of Zeta Distributions
نویسندگان
چکیده
Abstract The Riemann zeta distribution, defined as the one whose characteristic function is the normalised Riemann zeta function, is an interesting example of an infinitely divisible distribution. The infinite divisibility of the distribution has been proved with recourse to the Euler product of the Riemann zeta function. In this paper, we look at multiple zeta-star function, which is a multi-dimensional generalisation of the Riemann zeta function and is believed to have no Euler product, and show that the corresponding distribution is not infinitely divisible.
منابع مشابه
Modeling of Infinite Divisible Distributions Using Invariant and Equivariant Functions
Basu’s theorem is one of the most elegant results of classical statistics. Succinctly put, the theorem says: if T is a complete sufficient statistic for a family of probability measures, and V is an ancillary statistic, then T and V are independent. A very novel application of Basu’s theorem appears recently in proving the infinite divisibility of certain statistics. In addition ...
متن کاملFree infinite divisibility for beta distributions and related ones ∗
We prove that many of beta, beta prime, gamma, inverse gamma, Student tand ultraspherical distributions are freely infinitely divisible, but some of them are not. The latter negative result follows from a local property of probability density functions. Moreover, we show that the Gaussian, many of ultraspherical and Student t-distributions have free divisibility indicator 1.
متن کاملInfinitely Divisible Distributions for Rectangular Free Convolution: Classification and Matricial Interpretation
In a previous paper ([B-G1]), we defined the rectangular free convolution ⊞ λ . Here, we investigate the related notion of infinite divisibility, which happens to be closely related the classical infinite divisibility: there exists a bijection between the set of classical symmetric infinitely divisible distributions and the set of ⊞ λ -infinitely divisible distributions, which preserves limit t...
متن کاملOn the Non-classical Infinite Divisibility of Power Semicircle Distributions
The family of power semicircle distributions defined as normalized real powers of the semicircle density is considered. The marginals of uniform distributions on spheres in high-dimensional Euclidean spaces are included in this family and a boundary case is the classical Gaussian distribution. A review of some results including a genesis and the so-called Poincaré’s theorem is presented. The mo...
متن کاملOn infinite divisibility and embedding of probability measures on a locally compact Abelian group
In the present note the authors supplement significant properties of infinitely divisible and embeddable probability measures on a locally compact Abelian group G. There are at least two versions of infinite divisibility appearing in the literature which deserve special attention, and the problem of embedding those measures leads directly to the study of continuous convolution semigroups on G. ...
متن کامل