Small Ball Probabilities for Linear Images of High-Dimensional Distributions
Open Access Publications from the University of California

## Small Ball Probabilities for Linear Images of High-Dimensional Distributions

• Author(s): Rudelson, Mark;
• Vershynin, Roman
• et al.

## Published Web Location

https://doi.org/10.1093/imrn/rnu243
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.

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