Preservation of Infinite Divisibility under Mixing and Related Topics.

Steutel, F.W.

doi:10.2307/2556202

Cited by 32 publications

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“…of a mixture of exponential distributions), was proved earlier using two distinct approaches by Goldie (1967) and Steutel (1967), respectively, with the proof given by the former based implicitly on Kaluza's result. Steutel and van Harn (2004) and Sapatinas et al (2011) have unified the literature on Kaluza-Steutel and Goldie-Steutel results and have shed further light on aspects of the results of relevance to these, such as Theorem 2.3.1 of Steutel (1970); incidentally the latter theorem of Steutel referred to here implies, in view of the closure property of the class of i.d. distributions, that, if X and Y are independent random variables, with Y exponential and X real (not necessarily nonnegative), then XY (i.e.…”

Section: Introductionmentioning

confidence: 70%

Characterizations of Probability Distributions with Completely Monotone Hazard Functions

Albassam

2015

IJSP

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In this article, we characterize the classes of absolutely continuous distributions concentrated on (0, ∞) and discrete distributions concentrated on {0, 1, 2, ...}, with (non-vanishing survivor functions having) completely monotone hazard functions; in the latter case, we refer to the hazard functions also as the hazard sequences. These provide us with characterizations of the certain specialized versions of mixtures of exponential and geometric distributions with mixing distributions, satisfying some further criteria, which by the Goldie-Steutel theorem and a result of Kaluza are seen to be specialized versions of infinitely divisible distributions. We shed light on the implications of our findings, giving some pertinent examples and remarks.

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Section: Introductionmentioning

confidence: 70%

Characterizations of Probability Distributions with Completely Monotone Hazard Functions

Albassam

2015

IJSP

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“…From the techniques applied in (4.9)-(4.14), we examine and conclude that k 2 (t) is non-negative and bounded function for α > 1 and α 6 = 2, 3, ..., ∀t ∈ (−∞, ∞) and thus the density function g 2 (v) is infinitely divisible (See also [3], [8], [9], [10] [11]) and therefore the distribution with density function (1.9) is also infinitely divisible.…”

Section: On the Infinite Divisibilitymentioning

confidence: 99%

“…Several authors studied infinite divisibility of many functions and their related topics (See,, Bondesson [2], Goovaerts, D Hooge and Pril [3], Kelkar [5], Steutel [8], Takano [9,10] and Thorin [11] etc.). Motivated by the above work and from (1.1), (1.3) and (1.5), we introduce following equalities in the form of probability density functions:…”

Section: Introductionmentioning

confidence: 99%

On the Distributions of the Densities Involving Non-Zero Zeros of Bessel and Legendre Functions and Their Infinite Divisibility

Kumar¹,

Pathan

Chandel³

2010

Proyecciones (Antofagasta)

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In the present paper, we introduce the probability density functions involving non-zero zeros of the Bessel and Legendre functions. Then, we evaluate the distributions of the characteristic functions defined by these probability density functions and again obtain their related functions and polynomials. Finally, we prove the infinite divisibility of these probability density functions.2000 Mathematics Subject Classification : 33C20, 62E15, 60E05, 60E10.

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“…Let M£ = ME+*ME-. A representation of the Laplace transform of μ ^ ME+ is obtained by Steutel [5]. We state here his representation.…”

Section: Class Mementioning

confidence: 99%

On subclasses of infinitely divisible distributions on R related to hitting time distributions of 1-dimensional generalized diffusion processes

Yamazato

1992

Nagoya Mathematical Journal

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= Πί-i cn(s + ad-'/HU his + b,)'1 if m ^ 1.The author [8] shows that the upward first passage time distributions of birth and death processes belong to the class CME{. He [9] also shows that the class of distributions of hitting times of single points of generalized diffusion processes is a proper subclass of the closure CME+, in the weak convergence sense, of CMEl.Let CMEL be the class of distributions on R_ = (-°°, 0] whose mirror images belong to CME{. That is, μ e CMEL if and only if μ(du) = μ(-du) belongs toCMEί. Let CME f be the class of μ = μi * μ 2 with μι e CME{ and μ 2 e CMEL.

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Preservation of Infinite Divisibility under Mixing and Related Topics.

Cited by 32 publications

References 0 publications

Characterizations of Probability Distributions with Completely Monotone Hazard Functions

Characterizations of Probability Distributions with Completely Monotone Hazard Functions

On the Distributions of the Densities Involving Non-Zero Zeros of Bessel and Legendre Functions and Their Infinite Divisibility

On subclasses of infinitely divisible distributions on R related to hitting time distributions of 1-dimensional generalized diffusion processes

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