Infinite divisibility of random variables and their integer parts

1996 ◽  
Vol 28 (3) ◽  
pp. 271-278 ◽  
Author(s):  
Lennart Bondesson ◽  
Gundorph K. Kristiansen ◽  
Fred W. Steutel
Keyword(s):  
Random Variables ◽  
Infinite Divisibility

scholarly journals Two-faced families of non-commutative random variables having bi-free infinitely divisible distributions

2016 ◽  
Vol 27 (04) ◽  
pp. 1650037 ◽  
Author(s):  
Mingchu Gao
Keyword(s):  
Limit Theorem ◽  
Fock Space ◽  
Random Variables ◽  
Infinite Divisibility ◽  
Triangular Array ◽  
Divisible Distribution ◽  
Infinitely Divisible Distributions ◽  
Space Operator ◽  
Infinitely Divisible ◽  
Operator Model

We study two-faced families of non-commutative random variables having bi-free (additive) infinitely divisible distributions. We prove a limit theorem of the sums of bi-free two-faced families of random variables within a triangular array. As a corollary of our limit theorem, we get Voiculescu’s bi-free central limit theorem. Using the full Fock space operator model, we show that a two-faced pair of random variables has a bi-free (additive) infinitely divisible distribution if and only if its distribution is the limit distribution in our limit theorem. Finally, we characterize the bi-free (additive) infinite divisibility of the distribution of a two-faced pair of random variables in terms of bi-free Levy processes.

scholarly journals On the law of homogeneous stable functionals

2019 ◽  
Vol 23 ◽  
pp. 82-111
Author(s):  
Julien Letemplier ◽  
Thomas Simon
Keyword(s):  
Random Variables ◽  
Infinite Divisibility ◽  
Explicit Computation ◽  
Infinite Products ◽  
Stable Lévy Process ◽  
Distributional Properties ◽  
Lamperti Transformation ◽  
Beta Random Variables ◽  
Generalized Gamma Convolution ◽  
The Law

LetAbe theLq-functional of a stable Lévy process starting from one and killed when crossing zero. We observe thatAcan be represented as the independent quotient of two infinite products of renormalized Beta random variables. The proof relies on Markovian time change, the Lamperti transformation, and an explicit computation performed in [38] on perpetuities of hypergeometric Lévy processes. This representation allows us to retrieve several factorizations previously shown by various authors, and also to derive new ones. We emphasize the connections betweenAand more standard positive random variables. We also investigate the law of Riemannian integrals of stable subordinators. Finally, we derive several distributional properties ofArelated to infinite divisibility, self-decomposability, and the generalized Gamma convolution.

Some further results in infinite divisibility

1977 ◽  
Vol 82 (2) ◽  
pp. 289-295 ◽  
Author(s):  
D. N. Shanbhag ◽  
D. Pestana ◽  
M. Sreehari
Keyword(s):  
Positive Integer ◽  
Random Variables ◽  
Random Variable ◽  
Infinite Divisibility ◽  
Stable Laws ◽  
Infinitely Divisible ◽  
Stable Random Variable ◽  
Stable Random Variables ◽  
Image Position

Goldie (2), Steutel (8, 9), Kelker (4), Keilson and Steutel (3) and several others have studied the mixtures of certain distributions which are infinitely divisible. Recently Shanbhag and Sreehari (7) have proved that if Z is exponential with unit parameter and for 0 < α < 1, if Yx is a positive stable random variable with , t ≥ 0 and independent of Z, then for every 0 < α < 1Using this result, they have obtained several interesting results concerning stable random variables including some extensions of the results of the above authors. More recently, Williams (11) has used the same approach to show that if , where n is a positive integer ≥ 2, then is distributed as the product of n − 1 independent gamma random variables with index parameters α, 2α, …, (n − 1) α. Prior to these investigations, Zolotarev (12) had studied the problems of M-divisibility of stable laws.

Limit Theorems for Additive Conditionally Free Convolution

2011 ◽  
Vol 63 (1) ◽  
pp. 222-240 ◽  
Author(s):  
Jiun-Chau Wang
Keyword(s):  
Weak Convergence ◽  
Limit Theorems ◽  
Random Variables ◽  
Infinite Divisibility ◽  
Triangular Array ◽  
Free Convolution ◽  
Analytic Methods ◽  
Free Random Variables

AbstractIn this paper we determine the limiting distributional behavior for sums of infinitesimal conditionally free random variables. We show that the weak convergence of classical convolution and that of conditionally free convolution are equivalent for measures in an infinitesimal triangular array, where the measures may have unbounded support. Moreover, we use these limit theorems to study the conditionally free infinite divisibility. These results are obtained by complex analytic methods without reference to the combinatorics of c-free convolution.

Characteristic Functions

2021 ◽  
pp. 213-234
Author(s):  
James Davidson
Keyword(s):  
Number Theory ◽  
Characteristic Function ◽  
Complex Number ◽  
Random Variables ◽  
Infinite Divisibility ◽  
Characteristic Functions ◽  
Conditional Distributions ◽  
Inversion Theorem ◽  
Sums Of Random Variables

This chapter begins with a look at convolutions and the distribution of sums of random variables. It briefly surveys complex number theory before defining the characteristic function and studying its properties, with a range of examples. The concept of infinite divisibility is introduced. The important inversion theorem is treated and finally consideration is given to characteristic functions in conditional distributions.

Probability Functions which are Proportional to Characteristic Functions and the Infinite Divisibility of the von Mises Distribution

1975 ◽  
Vol 12 (S1) ◽  
pp. 19-28 ◽  
Author(s):  
Toby Lewis
Keyword(s):  
Continuous Distribution ◽  
Random Variables ◽  
Discrete Distribution ◽  
Infinite Divisibility ◽  
Reciprocal Relationship ◽  
Characteristic Functions ◽  
Von Mises Distribution ◽  
Infinitely Divisible ◽  
Von Mises ◽  
Divisibility Properties

Reciprocal pairs of continuous random variables on the line are considered, such that the density function of each is, to within a norming factor, the characteristic function of the other. The analogous reciprocal relationship between a discrete distribution on the line and a continuous distribution on the circle is also considered. A conjecture is made regarding infinite divisibility properties of such pairs of random variables. It is shown that the von Mises distribution is infinitely divisible for sufficiently small values of the concentration parameter.

Normal approximations to discrete unimodal distributions

1986 ◽  
Vol 23 (04) ◽  
pp. 1013-1018
Author(s):  
B. G. Quinn ◽  
H. L. MacGillivray
Keyword(s):  
Stationary Distribution ◽  
Sufficient Conditions ◽  
Random Variables ◽  
Hypergeometric Distribution ◽  
Death Process ◽  
Normal Approximations ◽  
Discrete Random Variables ◽  
Birth Death

Sufficient conditions are presented for the limiting normality of sequences of discrete random variables possessing unimodal distributions. The conditions are applied to obtain normal approximations directly for the hypergeometric distribution and the stationary distribution of a special birth-death process.

Sign in / Sign up

Export Citation Format

Share Document