An asymptotic thin shell condition and large deviations for random multidimensional projections

Large deviations for random matrices in the orthogonal group and Stiefel manifold with applications to random projections of product distributions

“…Moreover, an interesting connection between the study of large (and moderate) deviations for logconcave distributions and the famous Kannan-Lovász-Simonovits conjecture was established in [7]. Other than that a variety of large deviation results have been obtained in the last five years, among others, [24,41,42,43,46,50,51,52,57]. Beyond that, in [39] and the subsequent works [4,38], it has been demonstrated how ideas and methods from large deviation theory, such as the maximum entropy principle, its relation to Gibbs measures, and Gibbs conditioning, allow one to lift classical results for ℓ p -balls to more general symmetric Banach spaces (similar ideas have recently been used by Barthe and Wolff [12] studying Orlicz spaces).…”

Sanov-type large deviations and conditional limit theorems for high-dimensional Orlicz balls

“…Considering such Gibbs measures at suitable critical temperatures with potentials given by an Orlicz function allows us to work around the problem arising from the lack of a probabilistic representation. This idea has recently been put forward by Kabluchko and Prochno [12] (see particularly Subsection 1.2 there) in their probabilistic approach to the geometry of Orlicz balls and has also successfully been used in [1], [2], and [15].…”

“…Thus, as a composition of Fréchet-differentiable mappings, ϕ V (α(λ), λ) is Fréchet-differentiable. To see that α(λ) is the minimizer in (15), we note that by the Cauchy-Schwarz inequality, α → ϕ V (α, λ) is strictly convex and so is α → ϕ V (α, λ) − αR. Since ϕ V (α, λ) → ∞ as α → 0, we find an unique minimizer α(λ) ∈ (−∞, 0) with…”

“…This is necessary to introduce the exponential change of measure used in the proof of Proposition 3.11 in [15]. In [15] this condition may seem artificial at first, but looking at the proof of Theorem A, one sees that this condition appears in a quite natural way.…”

Thin-shell theory for rotationally invariant random simplices

“…Among others, there is Schmuckenschläger's central limit theorem related to the volume of intersections of n p -balls [52] and its multivariate version by Kabluchko, Prochno, and Thäle who also obtained moderate and large deviations principles [36,37]. Then there is the prominent central limit theorem for convex bodies proved by Klartag, showing that most lower-dimensional marginals of a random vector uniformly distributed in an isotropic convex body are approximately Gaussian [41], and a number of other results in which limit theorems related to analytic and geometric aspects of high-dimensional objects have been established [3,4,5,7,10,12,15,23,31,33,35,38,39,40,44,50,51,54,56,57].…”