scholarly journals Symmetric polynomials of random variables attracted to an infinitely divisible law

1986 ◽  
Vol 71 (4) ◽  
pp. 491-500 ◽  
Author(s):  
Florin Avram ◽  
Murad S. Taqqu
Keyword(s):  
Random Variables ◽  
Symmetric Polynomials ◽  
Infinitely Divisible ◽  
Infinitely Divisible Law

scholarly journals On Exact Sampling of Nonnegative Infinitely Divisible Random Variables

2012 ◽  
Vol 44 (3) ◽  
pp. 842-873 ◽  
Author(s):  
Zhiyi Chi
Keyword(s):  
Basic Result ◽  
Random Variables ◽  
Random Variable ◽  
Fixed Number ◽  
Rejection Sampling ◽  
Infinitely Divisible ◽  
Exact Sampling ◽  
Special Cases ◽  
Wide Range ◽  
Lévy Density

Nonnegative infinitely divisible (i.d.) random variables form an important class of random variables. However, when this type of random variable is specified via Lévy densities that have infinite integrals on (0, ∞), except for some special cases, exact sampling is unknown. We present a method that can sample a rather wide range of such i.d. random variables. A basic result is that, for any nonnegative i.d. random variable X with its Lévy density explicitly specified, if its distribution conditional on X ≤ r can be sampled exactly, where r > 0 is any fixed number, then X can be sampled exactly using rejection sampling, without knowing the explicit expression of the density of X. We show that variations of the result can be used to sample various nonnegative i.d. random variables.

A note on passage times and infinitely divisible distributions

1967 ◽  
Vol 4 (2) ◽  
pp. 402-405 ◽  
Author(s):  
H. D. Miller
Keyword(s):  
Random Walk ◽  
Markov Process ◽  
Continuous Time ◽  
Random Variables ◽  
Independent Random Variables ◽  
Infinitely Divisible Distributions ◽  
Infinitely Divisible ◽  
Time Intervals ◽  
Mutually Independent ◽  
Passage Times

Let X(t) be the position at time t of a particle undergoing a simple symmetrical random walk in continuous time, i.e. the particle starts at the origin at time t = 0 and at times T1, T1 + T2, … it undergoes jumps ξ1, ξ2, …, where the time intervals T1, T2, … between successive jumps are mutually independent random variables each following the exponential density e–t while the jumps, which are independent of the τi, are mutually independent random variables with the distribution . The process X(t) is clearly a Markov process whose state space is the set of all integers.

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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