scholarly journals Distributions with complete monotone derivative and geometric infinite divisibility

1990 ◽  
Vol 22 (3) ◽  
pp. 751-754 ◽  
Author(s):  
R. N. Pillai ◽  
E. Sandhya
Keyword(s):  
Infinite Divisibility ◽  
Infinitely Divisible Distributions ◽  
Infinitely Divisible ◽  
Proper Subclass ◽  
Geometric Infinite Divisibility

It is shown that a distribution with complete monotone derivative is geometrically infinitely divisible and that the class of distributions with complete monotone derivative is a proper subclass of the class of geometrically infinitely divisible distributions.

scholarly journals Distributions with complete monotone derivative and geometric infinite divisibility

1990 ◽  
Vol 22 (03) ◽  
pp. 751-754
Author(s):  
R. N. Pillai ◽  
E. Sandhya
Keyword(s):  
Infinite Divisibility ◽  
Infinitely Divisible Distributions ◽  
Infinitely Divisible ◽  
Proper Subclass ◽  
Geometric Infinite Divisibility

It is shown that a distribution with complete monotone derivative is geometrically infinitely divisible and that the class of distributions with complete monotone derivative is a proper subclass of the class of geometrically infinitely divisible distributions.

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.

Convolution equivalence and infinite divisibility

2004 ◽  
Vol 41 (02) ◽  
pp. 407-424 ◽  
Author(s):  
Anthony G. Pakes
Keyword(s):  
Distribution Function ◽  
Poisson Distribution ◽  
Infinite Divisibility ◽  
Compound Poisson Distribution ◽  
Infinitely Divisible Distributions ◽  
Random Sum ◽  
Lévy Measure ◽  
Infinitely Divisible ◽  
Tail Behaviour ◽  
Equivalence Result

Known results relating the tail behaviour of a compound Poisson distribution function to that of its Lévy measure when one of them is convolution equivalent are extended to general infinitely divisible distributions. A tail equivalence result is obtained for random sum distributions in which the summands have a two-sided distribution.

Geometric Infinite Divisibility and the Waiting Time Paradox

2005 ◽  
Vol 57 (1-2) ◽  
pp. 129-136
Author(s):  
R.N. Pillai ◽  
Saji Kumar V. R.
Keyword(s):  
Waiting Time ◽  
Renewal Process ◽  
Random Variable ◽  
Infinite Divisibility ◽  
Infinitely Divisible ◽  
Geometric Infinite Divisibility

It is shown that the waiting time W in a stationary renewal process generated by X has the form W = X+ Y, with Y non­negative independent of X if and only if X is a geometrically infinitely divisible random variable. This is an improvement over Van Harn and Steutel (1995) where the converse is left unproved .

Convolution equivalence and infinite divisibility

2004 ◽  
Vol 41 (2) ◽  
pp. 407-424 ◽  
Author(s):  
Anthony G. Pakes
Keyword(s):  
Distribution Function ◽  
Poisson Distribution ◽  
Infinite Divisibility ◽  
Compound Poisson Distribution ◽  
Infinitely Divisible Distributions ◽  
Random Sum ◽  
Lévy Measure ◽  
Infinitely Divisible ◽  
Tail Behaviour ◽  
Equivalence Result

Known results relating the tail behaviour of a compound Poisson distribution function to that of its Lévy measure when one of them is convolution equivalent are extended to general infinitely divisible distributions. A tail equivalence result is obtained for random sum distributions in which the summands have a two-sided distribution.

Poisson mixtures and quasi-infinite divisibility of distributions

1979 ◽  
Vol 16 (01) ◽  
pp. 138-153 ◽  
Author(s):  
Prem S. Puri ◽  
Charles M. Goldie
Keyword(s):  
Probability Distribution ◽  
Infinite Divisibility ◽  
Infinitely Divisible Distributions ◽  
Closure Properties ◽  
Infinitely Divisible ◽  
Poisson Mixture ◽  
Mixing Distribution ◽  
Poisson Mixtures ◽  
Mixing Distributions ◽  
Weakened Form

Any probability distribution on [0,∞) can function as the mixing distribution for a Poisson mixture, i.e. a mixture of Poisson distributions. The mixing distribution is called quasi-infinitely divisible (q.i.d.) if it renders the Poisson mixture infinitely divisible, or λ-q.i.d. if it does so after scaling by a factor λ> 0, or ∗-q.i.d. if it is λ-q.i.d. for some λ. These classes of distributions include the infinitely divisible distributions, and each exhibits many of the properties of the latter class but in weakened form. The paper presents the main properties of the classes and the class of Poisson mixtures, including characterisations of membership, relation with cumulants, and closure properties. Examples are given that establish among other things strict inclusions between the classes of mixing distributions.

Poisson mixtures and quasi-infinite divisibility of distributions

1979 ◽  
Vol 16 (1) ◽  
pp. 138-153 ◽  
Author(s):  
Prem S. Puri ◽  
Charles M. Goldie
Keyword(s):  
Probability Distribution ◽  
Infinite Divisibility ◽  
Infinitely Divisible Distributions ◽  
Closure Properties ◽  
Infinitely Divisible ◽  
Poisson Mixture ◽  
Mixing Distribution ◽  
Poisson Mixtures ◽  
Mixing Distributions ◽  
Weakened Form

Any probability distribution on [0,∞) can function as the mixing distribution for a Poisson mixture, i.e. a mixture of Poisson distributions. The mixing distribution is called quasi-infinitely divisible (q.i.d.) if it renders the Poisson mixture infinitely divisible, or λ-q.i.d. if it does so after scaling by a factor λ> 0, or ∗-q.i.d. if it is λ-q.i.d. for some λ. These classes of distributions include the infinitely divisible distributions, and each exhibits many of the properties of the latter class but in weakened form. The paper presents the main properties of the classes and the class of Poisson mixtures, including characterisations of membership, relation with cumulants, and closure properties. Examples are given that establish among other things strict inclusions between the classes of mixing distributions.

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.

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