Computationally tractable 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. As it only uses $q=\lfloor mp/n\rfloor$ coordinates of the observations in a subsample of size $m \ll n$ from the original data, it is computationally tractable for large scale data. In the high-dimensional scenario $p/n\rightarrow c\in (0,\infty)$, this fully nonparametric bootstrap is shown to consistently reproduce the empirical spectral measure if $m/n\rightarrow 0$. If $m^2/n\rightarrow 0$, 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.