Nonparametric bootstrap of high-dimensional sample covariance matrices

We introduce a new "$(m,mp/n)$ out of $(n,p)$" sampling-with-replace\-ment bootstrap for eigenvalue statistics of high-dimensional sample covariance matrices based on n independent p-dimensional random vectors. In the high-dimensional scenario p/n→ c∈ (0,∞), this fully nonparametric and computationally tractable bootstrap is shown to consistently reproduce the empirical spectral measure if m/n→ 0. If m²/n→ 0, it approximates correctly the distribution of linear spectral statistics. The crucial component is a suitably defined Representative Subpopulation Condition which is shown to be verified in a large variety of situations. Our proofs are conducted under minimal moment requirements and incorporate delicate results on non-centered quadratic forms, combinatorial trace moments estimates as well as a conditional bootstrap martingale CLT which may be of independent interest.