scholarly journals A Note on the Gaps in the Support of Discretely Infinitely Divisible Laws

2013 ◽  
Vol 2013 ◽  
pp. 1-4
Author(s):  
Anthony G. Pakes ◽  
S. Satheesh
Keyword(s):  
Real Line ◽  
Divisible Distribution ◽  
Infinitely Divisible Distributions ◽  
Infinitely Divisible Distribution ◽  
Infinitely Divisible ◽  
Compound Poisson ◽  
Infinitely Divisible Laws

We discuss the nature of gaps in the support of a discretely infinitely divisible distribution from the angle of compound Poisson laws/processes. The discussion is extended to infinitely divisible distributions on the nonnegative real line.

scholarly journals On convergence and compactness in variation with shift of discrete probability laws

2021 ◽  
Vol 8 (3) ◽  
pp. 385-393
Author(s):  
Ivan A. Alexeev ◽  
◽  
Alexey A. Khartov ◽  
Keyword(s):  
Real Line ◽  
Discrete Distribution ◽  
Distribution Functions ◽  
Divisible Distribution ◽  
Characteristic Functions ◽  
Infinitely Divisible Distribution ◽  
Infinitely Divisible ◽  
The Real ◽  
Convergence In Variation ◽  
Compactness Theorems

We consider a class of discrete distribution functions, whose characteristic functions are separated from zero, i. e. their absolute values are greater than positive constant on the real line. The class is rather wide, because it contains discrete infinitely divisible distribution functions, functions of lattice distributions, whose characteristic functions have no zeroes on the real line, and also distribution functions with a jump greater than 1/2. Recently the authors showed that characteristic functions of elements of this class admit the Lévy-Khinchine type representations with non-monotonic spectral function. Thus our class is included in the set of so called quasi-infinitely divisible distribution functions. Using these representation the authors also obtained limit and compactness theorems with convergence in variation for the sequences from this class. This note is devoted to similar results concerning convergence and compactness but with weakened convergence in variation. Replacing of type of convergence notably expands applicability of the results.

A Condition For Existence Of Moments Of Infinitely Divisible Distributions

1956 ◽  
Vol 8 ◽  
pp. 69-71 ◽  
Author(s):  
J. M. Shapiro
Keyword(s):  
Characteristic Function ◽  
Nondecreasing Function ◽  
Divisible Distribution ◽  
Infinitely Divisible Distributions ◽  
Real Constant ◽  
Infinitely Divisible Distribution ◽  
Infinitely Divisible ◽  
Image Position

Let F(x) be an infinitely divisible distribution and let ϕ(t) be its characteristic function. As is well known according to the formula of Lévy and Khintchine, ϕ(t) has the following representation:1where γ is a real constant and G(u) is a bounded nondecreasing function.

scholarly journals SECOND ORDER SUBEXPONENTIALITY AND INFINITE DIVISIBILITY

2020 ◽  
pp. 1-24 ◽  
Author(s):  
TOSHIRO WATANABE
Keyword(s):  
Asymptotic Behaviour ◽  
Real Line ◽  
Infinite Divisibility ◽  
Second Order ◽  
Divisible Distribution ◽  
Infinitely Divisible Distribution ◽  
Lévy Measure ◽  
Infinitely Divisible ◽  
Exponential Moment ◽  
The Real

We characterize the second order subexponentiality of an infinitely divisible distribution on the real line under an exponential moment assumption. We investigate the asymptotic behaviour of the difference between the tails of an infinitely divisible distribution and its Lévy measure. Moreover, we study the second order asymptotic behaviour of the tail of the $t$ th convolution power of an infinitely divisible distribution. The density version for a self-decomposable distribution on the real line without an exponential moment assumption is also given. Finally, the regularly varying case for a self-decomposable distribution on the half line is discussed.

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 Infinitely Divisible of Meixner Distribution

2018 ◽  
Vol 3 (4) ◽  
pp. 147
Author(s):  
Dodi Devianto ◽  
Jayanti Herli ◽  
Maiyastri Maiyastri ◽  
Rahma Diana Safitri
Keyword(s):  
Characteristic Function ◽  
Lévy Process ◽  
Goodness Of Fit ◽  
Divisible Distribution ◽  
Levy Process ◽  
Financial Data ◽  
Infinitely Divisible Distribution ◽  
Goodness Of Fit Test ◽  
Infinitely Divisible

The log-returns of most financial data show a significant leptokurtosis. For the better fit we showed a special levy process which is called the Meixner process. The Meixner distribution belongs to the class of infinitely divisible distribution chracterized by using characteristic function and it was proposed as a model for represented efficiently of the log-returns of financial data. The perfect fit of underlying Meixner distribution performing by using goodness of fit test.

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