A geometric characterization of indeterminate moment sequences

Merkes, E. P.; Wetzel, Marion

doi:10.2140/pjm.1976.65.409

Cited by 9 publications

(15 citation statements)

References 9 publications

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“…A Stieltjes moment sequence is indeterminate if and only if both the sequences ( ∆n ∆ 2,n−1 ) and ( ∆ 1,n ∆ 3,n−1 ) have non-zero limits as n → ∞ (cf. [18] Lemma 3). c).…”

Section: The Boolean Convolutionmentioning

confidence: 97%

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Measure Convolution Semigroups and Noninfinitely Divisible Probability Distributions

Wulfsohn

2005

J Math Sci

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Let µ be a probability measure (or corresponding random variable) such that all moments µ n exist. Knowledge of the moments is not sufficient to determine infinite divisibility of the measure; we show also that infinitely divisible, and in particular lognormal, distributions lose infinitely divisibilty when censored in certain ways even if all moments are arbitrarily close to those of the uncensored distribution. The moments of a composition of k copies of µ are expressed as combinatorial compositions of the µ n . We express the moments of the compositions in the context of occupancy problems, arranging n balls in k cells; the classical convolution is described by Maxwell-Boltzmann statistics and is multinomial. For certain non-infinitely divisible measures with moments increasing fast enough the indexing of a k-cell combinatorial composition is extended to indexing by non-negative real t and we construct classical convolution measure semigroups from amongst the t-indexed classes. We prove also that when a random variable with infinitely divisible distribution is embedded in a Lévy process (Y t ) then the t-indexed Maxwell-Boltzmann is the law of Y t . In order to get moment-based multinomial compositions indexed by a continuum we use random measures and random distributions rather than random variables. An alternative approach to embeddability of a non-infinitely divisible µ is by considering non-classical convolution measure semigroups; for example embedding µ in a Boolean convolution measure semigroup and retaining the multinomial character of the moments. Embedding µ in an Urbanik generalised convolution measure semigroup one loses the multinomial character of the moments.

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“…A Stieltjes moment sequence is indeterminate if and only if both the sequences ( ∆n ∆ 2,n−1 ) and ( ∆ 1,n ∆ 3,n−1 ) have non-zero limits as n → ∞ (cf. [18] Lemma 3). c).…”

Section: The Boolean Convolutionmentioning

confidence: 97%

“…When c 1 = µ 1 the sequence is determinate. By [18] Theorem 2, the sequence is indeterminate if and only if µ 1 > c 1 and (c 1 , µ 1 ] ⊂ D where D = m D m denotes the limit parabolic region as defined there. Proposition 4.13.…”

Section: The Boolean Convolutionmentioning

confidence: 99%

Measure Convolution Semigroups and Noninfinitely Divisible Probability Distributions

Wulfsohn

2005

J Math Sci

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“…As far as we are aware, there is no work, until now, giving such a presentation of most significant classical results on moment problems based on ideas and techniques similar to those used in this paper. We found a little strange that the paper by Merkes-Wetzel (1976) was somehow neglected for a long time. It is not in the list of references in papers and books written by leading specialists on the moment problem.…”

Section: About the Novelties In Our Approachmentioning

confidence: 99%

“…• The geometric interpretation of the indeterminacy conditions as developed by Merkes-Wetzel (1976).…”

Section: About the Novelties In Our Approachmentioning

confidence: 99%

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The Problem of Moments: A Bunch of Classical Results with Some Novelties

Novi Inverardi,

Tagliani,

Stoyanov

2023

Symmetry

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We summarize significant classical results on (in)determinacy of measures in terms of their finite positive integer order moments. Well known is the role of the smallest eigenvalues of Hankel matrices, starting from Hamburger’s results a century ago and ending with the great progress made only in recent times by C. Berg and collaborators. We describe here known results containing necessary and sufficient conditions for moment (in)determinacy in both Hamburger and Stieltjes moment problems. In our exposition, we follow an approach different from that commonly used. There are novelties well complementing the existing theory. Among them are: (a) to emphasize on the geometric interpretation of the indeterminacy conditions; (b) to exploit fine properties of the eigenvalues of perturbed symmetric matrices allowing to derive new lower bounds for the smallest eigenvalues of Hankel matrices (these bounds are used for concluding indeterminacy); (c) to provide new arguments to confirm classical results; (d) to give new numerical illustrations involving commonly used probability distributions.

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On Geometric Characterizations of an Indeterminate Stieltjes Moment Sequence

Merkes¹,

Wetzel²

1977

Pade and Rational Approximation

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A geometric characterization of indeterminate moment sequences

Cited by 9 publications

References 9 publications

Measure Convolution Semigroups and Noninfinitely Divisible Probability Distributions

Measure Convolution Semigroups and Noninfinitely Divisible Probability Distributions

The Problem of Moments: A Bunch of Classical Results with Some Novelties

On Geometric Characterizations of an Indeterminate Stieltjes Moment Sequence

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