A new look at random projections of the cube and general product measures
A new look at random projections of the cube and general product measures
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立方体随机投影和一般产品测量的新视角
DOI：
10.3150/20-bej1303
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发表时间：
2019
期刊：
影响因子：
--
通讯作者：
C. Thaele
中科院分区：
文献类型：
--
作者：
Z. Kabluchko;J. Prochno;C. Thaele
A strong law of large numbers for $d$-dimensional random projections of the $n$-dimensional cube is derived. It shows that with respect to the Hausdorff distance a properly normalized random projection of $[-1,1]^n$ onto $\mathbb{R}^d$ almost surely converges to a centered $d$-dimensional Euclidean ball of radius $\sqrt{2/\pi}$, as $n\to\infty$. For every point inside this ball we determine the asymptotic number of vertices and the volume of the part of the cube projected `close' to this point. Moreover, large deviations for random projections of general product measures are studied. Let $\nu^{\otimes n}$ be the $n$-fold product measure of a Borel probability measure $\nu$ on $\mathbb{R}$, and let $I$ be uniformly distributed on the Stiefel manifold of orthogonal $d$-frames in $\mathbb{R}^n$. It is shown that the sequence of random measures $\nu^{\otimes n}\circ(n^{-1/2}I^*)^{-1}$, $n\in\mathbb{N}$, satisfies a large deviations principle with probability $1$. The rate function is explicitly identified in terms of the moment generating function of $\nu$. At the heart of the proofs lies a transition trick which allows to replace the uniform projection by the Gaussian one. A number of concrete examples are discussed as well, including the uniform distributions on the cube $[-1,1]^n$ and the discrete cube $\{-1,1\}^n$ as a special cases.
A strong law of large numbers for $d$-dimensional random projections of the $n$-dimensional cube is derived. It shows that with respect to the Hausdorff distance a properly normalized random projection of $[-1,1]^n$ onto $\mathbb{R}^d$ almost surely converges to a centered $d$-dimensional Euclidean ball of radius $\sqrt{2/\pi}$, as $n\to\infty$. For every point inside this ball we determine the asymptotic number of vertices and the volume of the part of the cube projected `close' to this point. Moreover, large deviations for random projections of general product measures are studied. Let $\nu^{\otimes n}$ be the $n$-fold product measure of a Borel probability measure $\nu$ on $\mathbb{R}$, and let $I$ be uniformly distributed on the Stiefel manifold of orthogonal $d$-frames in $\mathbb{R}^n$. It is shown that the sequence of random measures $\nu^{\otimes n}\circ(n^{-1/2}I^*)^{-1}$, $n\in\mathbb{N}$, satisfies a large deviations principle with probability $1$. The rate function is explicitly identified in terms of the moment generating function of $\nu$. At the heart of the proofs lies a transition trick which allows to replace the uniform projection by the Gaussian one. A number of concrete examples are discussed as well, including the uniform distributions on the cube $[-1,1]^n$ and the discrete cube $\{-1,1\}^n$ as a special cases.
影子类的萨诺夫型大偏差
DOI：
10.1214/19-aihp989
发表时间：
2020
期刊：
Annales de l'Institut Henri Poincaré, Probabilités et Statistiques
影响因子：
--
作者：
Zakhar Kabluchko;Joscha Prochno;Christoph Thäle
通讯作者：
Christoph Thäle
âpn-balls II 中随机向量的高维极限定理
影响因子：
1.6
作者：
Zakhar Kabluchko;Joscha Prochno;Christoph Thäle
通讯作者：
Christoph Thäle
影响因子：
1
作者：
Kim, Steven S.;Ramanan, Kavita
通讯作者：
Ramanan, Kavita