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Finite sample properties of random covariance-type matrices have been thesubject of much research. In this paper we focus on the "lower tail" of such amatrix, and prove that it is subgaussian under a simple fourth momentassumption on the one-dimensional marginals of the random vectors. A similarresult holds for more general sums of random positive semidefinite matrices,and the (relatively simple) proof uses a variant of the so-called PAC-Bayesianmethod for bounding empirical processes.  We give two applications of the main result. In the first one we obtain a newfinite-sample bound for ordinary least squares estimator in linear regressionwith random design. Our result is model-free, requires fairly weak momentassumptions and is almost optimal. Our second application is to boundingrestricted eigenvalue constants of certain random ensembles with "heavy tails".These constants are important in the analysis of problems in Compressed Sensingand High Dimensional Statistics, where one recovers a sparse vector from asmall umber of linear measurements. Our result implies that heavy tails stillallow for the fast recovery rates found in efficient methods such as the LASSOand the Dantzig selector. Along the way we strengthen, with a fairly shortargument, a recent result of Rudelson and Zhou on the restricted eigenvalueproperty.

The lower tail of random quadratic forms, with applications to ordinary least squares and restricted eigenvalue properties