On the infinite divisibility of the product of two г-distributed stochastical variables

The authors give a positive answer to the question “if X_α is г-distributed of order α, and X_β of order β, with X_α and X_β independent, is X_αX_β infinitely divisible?”. This question, posed by Steutel in Ref. 1, has not been answered up to now, so far as they can find in the literature. In addition they show that the distribution of X_αX_β is a generalized г-distribution.     

Weak infinite powers of Blaschke products

Letb be a Blaschke product with zeros {z _n } in the open unit disk Δ. Let be the set of sequences of non-negative integersp=(p ₁,p ₂,…) such that ∑ _n=1 ^∞ p _n (1 − |z _n |) < ∞ andp _n →∞ asn→∞. We study the class of weak infinite powers ofb, Properties of these classes depend on the setS(b) of the cluster points in ∂Δ of {z _n }. It is proved thatS(b)=∂Δ if and only if , the Douglas algebra generated by . Also, it is proved thatdθ(S(b))=0 if and only if there exists an interpolating Blaschke productB such that .     

On the infinite divisibility of squared Gaussian processes

 We show that fractional Brownian motions with index in (0,1] satisfy a remarkable property: their squares are infinitely divisible. We also prove that a large class of Gaussian processes are sharing this property. This property then allows the construction of two-parameters families of processes having the additivity property of the squared Bessel processes. Received: 1 April 2002 / Revised version: 7 September 2002 / Published online: 19 December 2002 Mathematics Subject Classification (2000): 60E07, 60G15, 60J25, 60J55 Key words or phrases: Gaussian processes – Infinite divisibility – Markov processes     

On infinite divisibility of normed product of Cauchy densities

It is shown that the following probability density function

is an infinitely divisible density.     

Perfect powers in elliptic divisibility sequences

It is shown that there are finitely many perfect powers in an elliptic divisibility sequence whose first term is divisible by 2 or 3. For Mordell curves the same conclusion is shown to hold if the first term is greater than 1. Examples of Mordell curves and families of congruent number curves are given with corresponding elliptic divisibility sequences having no perfect power terms. The proofs combine primitive divisor results with modular methods for Diophantine equations.     

On the divisibility of the discriminant of an integral polynomial by prime powers

The discriminant of an integral polynomial is one of its main characteristics. It influences the distribution of its roots, the structure of the finite extension of the rational field generated by the polynomial's roots. In the paper, we show that, for any given prime power p ^b , there exists an irreducible polynomial with discriminant being a multiple of p ^b .     

Prime powers in elliptic divisibility sequences

Certain elliptic divisibility sequences are shown to contain only finitely many prime power terms. In some cases the methods prove that only finitely many terms are divisible by a bounded number of distinct primes.

     

Mean matrices and infinite divisibility

We consider matrices M with entries m_ij = m(λ_i, λ_j) where λ₁, … ,λ_n are positive numbers and m is a binary mean dominated by the geometric mean, and matrices W with entries w_ij = 1/m (λ_i, λ_j) where m is a binary mean that dominates the geometric mean. We show that these matrices are infinitely divisible for several much-studied classes of means.     

On infinite divisibility of positive definite functions arising from operator means

For a (point-wisely non-negative) positive definite function a certain criterion for its infinite divisibility (i.e., all its fractional powers are also positive definite) is obtained. This criterion enables us to show infinite divisibility for many positive definite functions appearing naturally in study of operator means. In particular, we determine when the function

     

Some recent results in infinite divisibility

In this paper, a survey is given of some recent developments in infinite divisibility. There are three main topics: (i) the occurrence of infinitely divisible distributions in applied stochastic processes such as queueing processes and birth-death processes, (ii) the construction of infinitely divisible distributions, mainly by mixing, and (iii) conditions for infinite divisibility in terms of distribution functions and densities.     

Transcendence of the values of infinite products in several variables

The aim of this paper is to prove the transcendence of certain infinite products. As applications, we get necessary and sufficient conditions for transcendence of the value of $\Pi_{k=0}^{\infty}(1+a_{k}^{(1)}{z_{1}r^{k}}+\cdot\cdot\cdot+a_{k}^{(m)}{z_{m}r^{k}})$ at appropriate algebraic points, where r ≥ 2 is an integer and {a_n ⁽ⁱ⁾}_{n≥ 0} (1 ≤ i ≤ m) are suitable sequences of algebraic numbers.     

On convolution powers on semidirect products

LetK be a compact group of linear operators of thed-dimensional spaceR ^d andG _K,d denote the semidirect productK byR ^d . It is shown that if an adapted probability measureμ onG _K,d is not scattered (i.e. for some compactF we havex ₀ ∈ R ^d (gF)>0), then there exists a nonzero vectorx ₀ ∈R ^d such thatk ₁(x ₀)=k ₂(x ₀) holds for all (k ₁,x ₁) and (k ₂,x ₂) belonging to the topological supportS(μ) of the measureμ. As a result we obtain that every adapted and strictly aperiodic probability measure on the group of all rigid motions of thed-dimensional Euclidian space is scattered. I thank the Foundation for Research Development for financial support.     

On the 4-dissections of certain infinite products

In this note, we give a generalization of Hirschhorn’s formulas on the 4-dissections of Ramanujan’s continued fraction R(q) and R ^?1(q) which were conjectured by Hirschhorn and proved by Lewis and Liu.     

A free analogue of Hincin's characterization of infinite divisibility

Hincin characterized the class of infinitely divisible distributions on the line as the class of all distributional limits of sums of infinitesimal independent random variables. We show that an analogue of this characterization is true in the addition theory of free random variables introduced by Voiculescu.