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 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.

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 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 The effect of random scale changes on limits of infinitesimal systems

1978 ◽  
Vol 1 (3) ◽  
pp. 339-372
Author(s):  
Patrick L. Brockett
Keyword(s):  
Sufficient Conditions ◽  
Random Variables ◽  
Domain Of Attraction ◽  
Divisible Distribution ◽  
Infinitely Divisible Distribution ◽  
Infinitely Divisible ◽  
Stable Law ◽  
Scale Parameters ◽  
Exact Relationship ◽  
Random Scale

SupposeS={{Xnj,   j=1,2,…,kn}}is an infinitesimal system of random variables whose centered sums converge in law to a (necessarily infinitely divisible) distribution with Levy representation determined by the triple(γ,σ2,M). If{Yj,   j=1,2,…}are independent indentically distributed random variables independent ofS, then the systemS′={{YjXnj,j=1,2,…,kn}}is obtained by randomizing the scale parameters inSaccording to the distribution ofY1. We give sufficient conditions on the distribution ofYin terms of an index of convergence ofS, to insure that centered sums fromS′be convergent. If such sums converge to a distribution determined by(γ′,(σ′)2,Λ), then the exact relationship between(γ,σ2,M)and(γ′,(σ′)2,Λ)is established. Also investigated is when limit distributions fromSandS′are of the same type, and conditions insuring products of random variables belong to the domain of attraction of a stable law.

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