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- Title
Geometrical Bounds for Variance and Recentered Moments.
- Authors
Lim, Tongseok; McCann, Robert J.
- Abstract
We bound the variance and other moments of a random vector based on the range of its realizations, thus generalizing inequalities of Popoviciu and of Bhatia and Davis concerning measures on the line to several dimensions. This is done using convex duality and (infinite-dimensional) linear programming. The following consequence of our bounds exhibits symmetry breaking, provides a new proof of Jung's theorem, and turns out to have applications to the aggregation dynamics modelling attractive–repulsive interactions: among probability measures on R n whose support has diameter at most 2 , we show that the variance around the mean is maximized precisely by those measures that assign mass 1 / (n + 1) to each vertex of a standard simplex. For 1 ≤ p < ∞ , the p th moment—optimally centered—is maximized by the same measures among those satisfying the diameter constraint.
- Subjects
DAVIS (Calif.); JUNG, C. G. (Carl Gustav), 1875-1961; LINEAR programming; PROBABILITY measures; SYMMETRY breaking
- Publication
Mathematics of Operations Research, 2022, Vol 47, Issue 1, p286
- ISSN
0364-765X
- Publication type
Article
- DOI
10.1287/moor.2021.1125