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

Author: Philipp J. di Dio
Journal: Proc. Amer. Math. Soc. 147 (2019), 3021-3038
MSC (2010): Primary 44A60, 14P10
Published electronically: March 21, 2019
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Abstract: We study truncated moment sequences of distribution mixtures, especially from Gaussian and log-normal distributions and their Carathéodory numbers. For $ \mathsf {A} = \{a_1,\dots ,a_m\}$ continuous (sufficiently differentiable) functions on $ \mathds {R}^n$ we give a general upper bound of $ m-1$ and a general lower bound of $ \left \lceil \frac {2m}{(n+1)(n+2)}\right \rceil $. 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 [J. Math. Anal. Appl. 461 (2018), pp. 1606-1638]. Therefore, for univariate polynomials $ \{1,x,\dots ,x^d\}$ at most $ \left \lceil \frac {d+1}{2}\right \rceil $ distributions are needed. For bivariate polynomials of degree at most $ 2d-1$ we find that $ \frac {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\}$ three 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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Philipp J. di Dio
Affiliation: Mathematisches Institut, Universität Leipzig, Augustusplatz 10/11, D-04109 Leipzig, Germany; Max Planck Institute for Mathematics in the Sciences, Inselstraße 22, D-04103 Leipzig, Germany

Keywords: Truncated moment problem, Carath\'eodory number, Gaussian, log-normal, finite mixture models, moment method
Received by editor(s): May 8, 2018
Received by editor(s) in revised form: July 19, 2018, and October 13, 2018
Published electronically: March 21, 2019
Additional Notes: The author was supported by the Deutsche Forschungsgemeinschaft (SCHM1009/6-1).
Communicated by: David Levin
Article copyright: © Copyright 2019 American Mathematical Society