×

Sharp lower bounds on the least singular value of a random matrix without the fourth moment condition. (English) Zbl 1318.60009

Summary: We obtain non-asymptotic lower bounds on the least singular value of \({\mathbf X}_{pn}^\top/\sqrt{n}\), where \({\mathbf X}_{pn}\) is a \(p\times n\) random matrix whose columns are independent copies of an isotropic random vector \(X_p\) in \( {\mathbb R}^p\). We assume that there exist \(M>0\) and \(\alpha\in(0,2]\) such that \({\mathbb P}(|(X_p,v)|>t)\leqslant M/t^{2+\alpha}\) for all \(t>0\) and any unit vector \(v\in{\mathbb R}^p\). These bounds depend on \(y=p/n\), \(\alpha\), \(M\) and are asymptotically optimal up to a constant factor.

MSC:

60B20 Random matrices (probabilistic aspects)
PDFBibTeX XMLCite
Full Text: DOI