Mixtures of infinitely divisible distributions as limit laws for sums of dependent random variables

1980 ◽  
Vol 51 (1) ◽  
pp. 101-113 ◽  
Author(s):  
Andrzej Kłopotowski
Keyword(s):  
Random Variables ◽  
Infinitely Divisible Distributions ◽  
Infinitely Divisible ◽  
Limit Laws ◽  
Dependent 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.

A remark on the central limit question for dependent random variables

1980 ◽  
Vol 17 (1) ◽  
pp. 94-101 ◽  
Author(s):  
Richard C. Bradley
Keyword(s):  
Central Limit ◽  
Random Variables ◽  
Stationary Sequence ◽  
Stable Limit ◽  
Limit Laws ◽  
Dependent Random Variables ◽  
Asymptotically Normal ◽  
Second Moments

Given a strictly stationary sequence {Xk, k = …, −1,0,1, …} of r.v.'s one defines for n = 1, 2, 3 …, . Here an example of {Xk} is given with finite second moments, for which Var(X1 + … + Xn)→∞ and ρ n → 0 as n→∞, but (X1 + … + Xn) fails to be asymptotically normal; instead there is partial attraction to non-stable limit laws.

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.

A remark on the central limit question for dependent random variables

1980 ◽  
Vol 17 (01) ◽  
pp. 94-101 ◽  
Author(s):  
Richard C. Bradley
Keyword(s):  
Central Limit ◽  
Random Variables ◽  
Stationary Sequence ◽  
Stable Limit ◽  
Limit Laws ◽  
Dependent Random Variables ◽  
Asymptotically Normal ◽  
Second Moments

Given a strictly stationary sequence {Xk, k= …, −1,0,1, …} of r.v.'s one defines forn= 1, 2, 3 …,. Here an example of {Xk} is given with finite second moments, for which Var(X1+ … +Xn)→∞ andρn→ 0 asn→∞, but (X1+ … +Xn) fails to be asymptotically normal; instead there is partial attraction to non-stable limit laws.

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