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Register a new userThin-shell concentration for random vectors in Orlicz balls via moderate deviations and Gibbs measures
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In this paper, we study the asymptotic thin-shell width concentration for random vectors uniformly distributed in Orlicz balls. We provide both asymptotic upper and lower bounds on the probability of such a random vector $X_n$ being in a thin shell of radius $sqrt{n}$ times the asymptotic value of $n^{-1/2}left(mathbb Eleft[| X_n|_2^2right]right)^{1/2}$ (as $ntoinfty$), showing that in certain ranges our estimates are optimal. In particular, our estimates significantly improve upon the currently best known general Lee-Vempala bound when the deviation parameter $t=t_n$ goes down to zero as the dimension $n$ of the ambient space increases. We shall also determine in this work the precise asymptotic value of the isotropic constant for Orlicz balls. Our approach is based on moderate deviation principles and a connection between the uniform distribution on Orlicz balls and Gibbs measures at certain critical inverse temperatures with potentials given by Orlicz functions, an idea recently presented by Kabluchko and Prochno in [The maximum entropy principle and volumetric properties of Orlicz balls, J. Math. Anal. Appl. {bf 495}(1) 2021, 1--19].
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Consider the projection of an $n$-dimensional random vector onto a random $k_n$-dimensional basis, $k_n leq n$, drawn uniformly from the Haar measure on the Stiefel manifold of orthonormal $k_n$-frames in $mathbb{R}^n$, in three different asymptotic regimes as $n rightarrow infty$: constant ($k_n=k$), sublinear ($k_n rightarrow infty$ but $k_n/n rightarrow 0$) and linear $k_n/n rightarrow lambda$ with $0 < lambda le 1$). When the sequence of random vectors satisfies a certain asymptotic thin shell condition, we establish annealed large deviation principles (LDPs) for the corresponding sequence of random projections in the constant regime, and for the sequence of empirical measures of the coordinates of the random projections in the sublinear and linear regimes. We also establish LDPs for certain scaled $ell_q$ norms of the random projections in these different regimes. Moreover, we verify our assumptions for various sequences of random vectors of interest, including those distributed according to Gibbs measures with superquadratic interaction potential, or the uniform measure on suitably scaled $ell_p^n$ balls, for $p in [1,infty)$, and generalized Orlicz balls defined via a superquadratic function. Our results complement the central limit theorem for convex sets and related results which are known to hold under a thin shell condition. These results also substantially extend existing large deviation results for random projections, which are first, restricted to the setting of measures on $ell_p^n$ balls, and secondly, limited to univariate LDPs (i.e., in $mathbb{R}$) involving either the norm of a $k_n$-dimensional projection or the projection of $X^{(n)}$ onto a random one-dimensional subspace. Random projections of high-dimensional random vectors are of interest in a range of fields including asymptotic convex geometry and high-dimensional statistics.
In this article we prove three fundamental types of limit theorems for the $q$-norm of random vectors chosen at random in an $ell_p^n$-ball in high dimensions. We obtain a central limit theorem, a moderate deviations as well as a large deviations principle when the underlying distribution of the random vectors belongs to a general class introduced by Barthe, Guedon, Mendelson, and Naor. It includes the normalized volume and the cone probability measure as well as projections of these measures as special cases. Two new applications to random and non-random projections of $ell_p^n$-balls to lower-dimensional subspaces are discussed as well. The text is a continuation of [Kabluchko, Prochno, Thale: High-dimensional limit theorems for random vectors in $ell_p^n$-balls, Commun. Contemp. Math. (2019)].
In [A dozen de {F}inetti-style results in search of a theory, Ann. Inst. H. Poincar{e} Probab. Statist. 23(2)(1987), 397--423], Diaconis and Freedman studied low-dimensional projections of random vectors from the Euclidean unit sphere and the simplex in high dimensions, noting that the individual coordinates of these random vectors look like Gaussian and exponential random variables respectively. In subsequent works, Rachev and Ruschendorf and Naor and Romik unified these results by establishing a connection between $ell_p^N$ balls and a $p$-generalized Gaussian distribution. In this paper, we study similar questions in a significantly generalized and unifying setting, looking at low-dimensional projections of random vectors uniformly distributed on sets of the form [B_{phi,t}^N := Big{(s_1,ldots,s_N)inmathbb{R}^N : sum_{ i =1}^Nphi(s_i)leq t NBig},] where $phi:mathbb{R}to [0,infty]$ is a potential (including the case of Orlicz functions). Our method is different from both Rachev-Ruschendorf and Naor-Romik, based on a large deviation perspective in the form of quantitati
Accurate estimation of tail probabilities of projections of high-dimensional probability measures is of relevance in high-dimensional statistics and asymptotic geometric analysis. For fixed $p in (1,infty)$, let $(X^{(n,p)})$ and $(theta^n)$ be independent sequences of random vectors with $theta^n$ distributed according to the normalized cone measure on the unit $ell_2^n$ sphere, and $X^{(n,p)}$ distributed according to the normalized cone measure on the unit $ell_p^n$ sphere. For almost every sequence of projection directions $(theta^n)$, (quenched) sharp large deviation estimates are established for suitably normalized (scalar) projections of $X^{n,p}$ onto $theta^n$, that are asymptotically exact (as the dimension $n$ tends to infinity). Furthermore, the case when $(X^{(n,p)})$ is replaced with $(mathscr{X}^{(n,p)})$, where $mathscr{X}^{(n,p)}$ is distributed according to the uniform (or normalized volume) measure on the unit $ell_p^n$ ball, is also considered. In both cases, in contrast to the (quenched) large deviation rate function, the prefactor exhibits a dependence on the projection directions $(theta^n)$ that encodes geometric information. Moreover, although the (quenched) large deviation rate functions for the sequences of random projections of $(X^{(n,p)})$ and $(mathscr{X}^{(n,p)})$ are known to coincide, it is shown that the prefactor distinguishes between these two cases. The results on the one hand provide quantitative estimates of tail probabilities of random projections of $ell_p^n$ balls and spheres, valid for finite $n$, generalizing previous results due to Gantert, Kim and Ramanan, and on the other hand, generalize classical sharp large deviation estimates in the spirit of Bahadur and Ranga Rao to a geometric setting.
The convex hull generated by the restriction to the unit ball of a stationary Poisson point process in the $d$-dimensional Euclidean space is considered. By establishing sharp bounds on cumulants, exponential estimates for large deviation probabilities are derived and the relative error in the central limit theorem on a logarithmic scale is investigated for a large class of key geometric characteristics. This includes the number of lower-dimensional faces and the intrinsic volumes of the random polytopes. Furthermore, moderate deviation principles for the spatial empirical measures induced by these functionals are also established using the method of cumulants. The results are applied to deduce, by duality, fine probabilistic estimates and moderate deviation principles for combinatorial parameters of a class of zero cells associated with Poisson hyperplane mosaics. As a special case this comprises the typical Poisson-Voronoi cell conditioned on having large inradius.
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