2018/04/30 by Philipp J. di Dio · 6 citations
Mathematics · #Advanced Combinatorial Mathematics #Bounded function #Gaussian #Gaussian random field #Mathematical functions and polynomials #Moment (physics) #Moment problem #Orthogonal polynomials #Polynomial #Random Matrices and Applications #Sequence (biology) #Univariate #Upper and lower bounds #math.PR #msc:14P10 #msc:44A60
paper · pdf · doi:10.1090/proc/14499
published in Proceedings of the American Mathematical Society 147(7), 3021-3038 (American Mathematical Society)
openalex created_date 2018/04/24 · arxiv created 2018/07/19 · openalex publication_date 2018/12/17 · arxiv updated 2019/07/01 · openalex updated_date 2026/08/05
We study truncated moment sequences of distribution mixtures, especially from Gaussian and log-normal distributions and their Carathéodory numbers. For \mathsf A = \a1,… ,am\ continuous (sufficiently differentiable) functions on \mathbb Rn we give a general upper bound of m-1 and a general lower bound of \lceil \frac 2m(n+1)(n+2) \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. 1606â1638]. Therefore, for univariate polynomials \1,x,… ,xd\ at most \lceil \frac d+12 \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,x2,x3,x5,x6\ 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.