Abstract
We derive tight lower bounds on the smallest eigenvalue of a sample covariance matrix of a centred isotropic random vector under weak or no assumptions on its components.
Highlights
We provide tight lower bounds on the smallest eigenvalue of a sample covariance matrix of a centred isotropic random vector under weak or no assumptions on its components
Lower bounds on the smallest eigenvalue of a sample covariance matrix play a crucial role in the least squares problems in high-dimensional statistics
For a random vector Xp in Rp, consider a random p × n matrix Xpn with independent columns {Xpk}nk=1 distributed as Xp and the Gram matrix n
Summary
Lower bounds on the smallest eigenvalue of a sample covariance matrix (or a Gram matrix) play a crucial role in the least squares problems in high-dimensional statistics (see, for example, [5]). For a random vector Xp in Rp, consider a random p × n matrix Xpn with independent columns {Xpk}nk=1 distributed as Xp and the Gram matrix n. If Xp is centred, n−1XpnXpn is the sample covariance matrix corresponding to the random sample {Xpk}nk=1. In this paper we derive sharp lower bounds for λp(n−1XpnXpn), where λp(A) is the smallest eigenvalue of a p × p matrix A. In proofs we use the same strategy as in [6]
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