Two Novel Characterizations of Self-Decomposability on the Half-Line

Mai, Jan-Frederik; Schenk, Steffen; Scherer, Matthias

doi:10.1007/s10959-015-0644-6

Cited by 9 publications

(12 citation statements)

References 24 publications

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“…The latter is precisely what is stated in Lemma 3.1 in [11] in case Ψ(0) = 1, the minimality condition was somehow occulted.…”

Section: Complete Monotonicity Property Of Functions Is Recognized By...supporting

confidence: 65%

“…Remark 9. The main contribution in [11] consists in Theorem 1.1 where it was stated in case Φ(0) = 0: Φ is a self-decomposable Bernstein function if and only if (Φ(xk) − Φ(yk)) k≥0 is completely alternating for every x > y > 0.…”

Section: ) φ Is a Self-decomposable Bernstein Function If And Only If...mentioning

confidence: 99%

“…Date: February 17, 2016. We prove that under natural regularity assumptions on f , the answer is affirmative for the first question (and then infirmative for the second) and this constitutes a kind of converse of Hausdorff's moment characterization theorem [13]. Mai, Schenk and Scherer [11] adapted a Widder's result [19] and used a specific technique from Copula theory in order to state, in their Lemma 3.1 and Theorem 1.1, that:…”

Section: Introductionmentioning

confidence: 95%

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Complete monotonicity and bernstein properties of functions are characterized by their restriction on N

Aguech,

Jedidi

2015

Preprint

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We give several new characterizations of completely monotone functions (CM) and Bernstein functions (BF) via two approaches: the first one is driven algebraically via elementary preserving mappings and the second one is developed in terms of the behavior of their restriction on N 0 . We give a complete answer to the following question: Can we affirm that a function f is completely monotone (resp. a Bernstein function) if we know that the sequence (f (k)) k is completely monotone (resp. alternating)? This approach constitutes a kind of converse of Hausdorff's moment characterization theorem in the context of completely monotone sequences.

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“…The latter is precisely what is stated in Lemma 3.1 in [11] in case Ψ(0) = 1, the minimality condition was somehow occulted.…”

Section: Complete Monotonicity Property Of Functions Is Recognized By...supporting

confidence: 65%

Section: ) φ Is a Self-decomposable Bernstein Function If And Only If...mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 95%

See 1 more Smart Citation

Complete monotonicity and bernstein properties of functions are characterized by their restriction on N

Aguech,

Jedidi

2015

Preprint

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“…for ψ : [0, ∞) → (0, 1], ψ(x) := exp(−Ψ 1 (x)). The resulting class C Λ,d is analytically tractable and is analyzed in detail and illustrated in [16]. As a concluding example, consider the self-decomposable Bernstein function…”

Section: Examples Of Tractable Familiesmentioning

confidence: 99%

Exchangeable exogenous shock models

Mai¹,

Schenk²,

Scherer³

2016

Bernoulli

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We characterize a comprehensive family of d-variate exogenous shock models. Analytically, we consider a family of multivariate distribution functions that arises from ordering, idiosyncratically distorting, and finally multiplying the arguments. Necessary and sufficient conditions on the involved distortions to yield a multivariate distribution function are given. Probabilistically, the attainable set of distribution functions corresponds to a large class of exchangeable exogenous shock models. Besides, the vector of exceedance times of an increasing additive stochastic process across independent exponential trigger variables is shown to constitute an interesting subclass of the considered distributions and yields a second probabilistic model. The alternative construction is illustrated in terms of two examples.

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“…Necessity, i.e. that self-decomposability can actually be characterized in terms of the multivariate survival functions (42), is shown in [62] and relies on some purely analytical, technical computations.…”

Section: Proofmentioning

confidence: 99%

The infinite extendibility problem for exchangeable real-valued random vectors

Mai

2019

Preprint

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We survey known solutions to the infinite extendibility problem for (necessarily exchangeable) probability laws on R d , which is: Can a given random vector X = (X 1 , . . . , X d ) be represented in distribution as the first d members of an infinite exchangeable sequence of random variables? This is the case if and only if X has a stochastic representation that is "conditionally iid" according to the seminal De Finetti's Theorem. Of particular interest are cases in which the original motivation behind the model X is not one of conditional independence. After an introduction and some general theory, the survey covers the traditional cases when X takes values in {0, 1} d , has a spherical law, a law with 1 -norm symmetric survival function, or a law with ∞ -norm symmetric density. The solutions in all these cases constitute analytical characterizations of mixtures of iid sequences drawn from popular, one-parametric probability laws on R, like the Bernoulli, the normal, the exponential, or the uniform distribution. The survey further covers the less traditional cases when X has a Marshall-Olkin distribution, a multivariate wide-sense geometric distribution, a multivariate extreme-value distribution, or is defined as a certain exogenous shock model including the special case when its components are samples from a Dirichlet prior. The solutions in these cases correspond to iid sequences drawn from random distribution functions defined in terms of popular families of non-decreasing stochastic processes, like a Lévy subordinator, a random walk, a process that is strongly infinitely divisible with respect to time, or an additive process. The survey finishes with a list of potentially interesting open problems. In comparison to former literature on the topic, this survey purposely dispenses with generalizations to the related and larger concept of finite exchangeability or to more general state spaces than R. Instead, it aims to constitute an up-to-date comprehensive collection of known and compelling solutions of the real-valued extendibility problem, accessible for both applied and theoretical probabilists, presented in a lecture-like fashion.

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Two Novel Characterizations of Self-Decomposability on the Half-Line

Cited by 9 publications

References 24 publications

Complete monotonicity and bernstein properties of functions are characterized by their restriction on N

Complete monotonicity and bernstein properties of functions are characterized by their restriction on N

Exchangeable exogenous shock models

The infinite extendibility problem for exchangeable real-valued random vectors

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