An asymptotic thin shell condition and large deviations for random multidimensional projections
نویسندگان
چکیده
It is well known that fluctuations of marginals high-dimensional random vectors satisfy a certain concentration estimate called the thin shell condition are approximately Gaussian. In this article we identify general on sequence under which one can exponential decay rate large deviation probabilities corresponding marginals. More precisely, consider projection an n-dimensional vector onto kn-dimensional basis, kn≤n, drawn uniformly from Haar measure Stiefel manifold orthonormal kn-frames in Rn, three different asymptotic regimes as n→∞: “constant” (kn=k), “sublinear” (kn→∞ but kn/n→0) and “linear” (kn/n→λ with 0<λ≤1). When satisfies “asymptotic condition”, establish principles for projections constant regime, empirical measures coordinates sublinear linear regimes. We also scaled ℓq norms all Moreover, show holds various sequences interest, including uniform suitably ℓpn balls, p∈[1,∞), generalized Orlicz balls defined via superquadratic function, class Gibbs interaction potential. Along way, obtain logarithmic asymptotics volumes may be independent interest. Euclidean multi-dimensional when p∈[1,2), exhibits unexpected phase transition thus disproving earlier conjecture due to Alonso-Gutiérrez et al. Random interest range fields convex geometry statistics.
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ژورنال
عنوان ژورنال: Advances in Applied Mathematics
سال: 2022
ISSN: ['1090-2074', '0196-8858']
DOI: https://doi.org/10.1016/j.aam.2021.102306