Small Ball Probabilities for Linear Images of High-Dimensional Distributions
Small Ball Probabilities for Linear Images of High-Dimensional Distributions
复制标题
高维分布线性图像的小球概率
DOI：
10.1093/imrn/rnu243
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发表时间：
2014
影响因子：
1
通讯作者：
R. Vershynin
中科院分区：
文献类型：
--
作者：
M. Rudelson;R. Vershynin
作者：鲁德森，马克；韦尔辛，罗马|摘要：我们研究表单$ ax $的随机向量的浓度属性，其中$ x =（x_1，...，x_n）$具有独立的坐标，$ a $是给定的矩阵。我们表明，每当$ x_i $的分布在线上散布时，$ ax $的分布在太空中会很好地传播。具体而言，假设$ x_i $以任何长度$ t $的任何给定间隔最多是$ p $的可能性。然后，$ ax $属于任何给定半径的球$ t \ | a \ | _ {hs} $的可能性最多是$（cp）^{0.9 r（a）} $，其中$ r（a）$表示$ a $ a和$ c $的稳定等级是绝对常数。
Author(s): Rudelson, Mark; Vershynin, Roman | Abstract: We study concentration properties of random vectors of the form $AX$, where $X = (X_1, ..., X_n)$ has independent coordinates and $A$ is a given matrix. We show that the distribution of $AX$ is well spread in space whenever the distributions of $X_i$ are well spread on the line. Specifically, assume that the probability that $X_i$ falls in any given interval of length $T$ is at most $p$. Then the probability that $AX$ falls in any given ball of radius $T \|A\|_{HS}$ is at most $(Cp)^{0.9 r(A)}$, where $r(A)$ denotes the stable rank of $A$ and $C$ is an absolute constant.