## Abstract

Given a random vector X in R^{n}, set X_{1},·, X_{N} to be independent copies of X and let Γ = 1/√N∑_{i}=1^{N} (X_{i}, ) e_{i} be the matrix whose rows are X_{1}√ N,·, X_{N}√N. We obtain new probabilistic lower bounds on the smallest singular value λ_{min}(Γ) in a rather general situation, and in particular, under the assumption that X is an isotropic random vector for which ⊃_{t}\in S^{n-1}P t, X|u L/u^{2+}η for some L,η >0. Our results imply that a Bai-Yin-type lower bound holds for η >2, and, up to a log-factor, for η =2 as well. The bounds hold without any additional assumptions on the Euclidean norm |X|ℓ_{2}^{n}. Moreover, we establish a nontrivial lower bound even without any higher moment assumptions (corresponding to the case η =0), if the linear forms satisfy a weak "small-ball" property. These estimates follow from general lower bounds on the infimum of the quadratic empirical process f →N^{-1}∑_{i}=1^{N} f^{2}(X_{i}) which are of independent interest.

Original language | English |
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Pages (from-to) | 12991-13008 |

Number of pages | 18 |

Journal | International Mathematics Research Notices |

Volume | 2015 |

Issue number | 23 |

DOIs | |

Publication status | Published - 2015 |

Externally published | Yes |