Multidimensional Truncated Moment Problems: Existence

Schmüdgen, Konrad

doi:10.1007/978-3-319-64546-9_17

Cited by 1 publication

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References 13 publications

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“…The theory and application of moments is rich, see e.g. [KS53], [Ric57], [Rog58], [AK62], [Akh65], [Kem68], [KN77], [Sch91], [Mat92], [Rez92], [CF96a], [CF96b], [Sim98], [CF00], [Sch03], [FP05], [CF05], [PS08], [Mar08], [Lau09], [FN10], [CF13], [Las15], [Sch15], [Sto16], [Fia17], [IKLS17], [SdD17], [Sch17], [RS18], [dDS18a], [dDS18b], and references therein. But in the present section we only present definitions and results needed in the following sections, especially from [dDS18a] and [dDS18b] with extensions to mixtures as presented in the introduction.…”

Section: Preliminariesmentioning

confidence: 99%

The multidimensional truncated moment problem: Gaussian and log-normal mixtures, their Carathéodory numbers, and set of atoms

Dio

2019

Proc. Amer. Math. Soc.

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We study truncated moment sequences of distribution mixtures, especially from Gaussian and log-normal distributions and their Carathéodory numbers. For A = {a 1 , . . . , am} continuous (sufficiently differentiable) functions on R n we give a general upper bound of m − 1 and a general lower bound of 2m (n+1)(n+2) . For polynomials of degree at most d in n variables we find that the number of Gaussian and log-normal mixtures is bounded by the Carathéodory numbers in [dDS18b]. Therefore, for univariate polynomials {1, x, . . . , x d } at most d+1 2 distributions are needed. For bivariate polynomials of degree at most 2d − 1 we find that 3d(d−1) 2 + 1 Gaussian distributions are sufficient. We also treat polynomial systems with gaps and find, e.g., that for {1, x 2 , x 3 , x 5 , x 6 } 3 Gaussian distributions are enough for almost all truncated moment sequences. For log-normal distributions the number is bounded by half of the moment number. We give an example of continuous functions where more Gaussian distributions are needed than Dirac delta measures. We show that any inner truncated moment sequence has a mixture which contains any given distribution.

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Section: Preliminariesmentioning

confidence: 99%

The multidimensional truncated moment problem: Gaussian and log-normal mixtures, their Carathéodory numbers, and set of atoms

Dio

2019

Proc. Amer. Math. Soc.

View full text Add to dashboard Cite

show abstract

Multidimensional Truncated Moment Problems: Existence

Cited by 1 publication

References 13 publications

The multidimensional truncated moment problem: Gaussian and log-normal mixtures, their Carathéodory numbers, and set of atoms

The multidimensional truncated moment problem: Gaussian and log-normal mixtures, their Carathéodory numbers, and set of atoms

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