Xn be i.i.d. sample in R p with zero mean and the covariance matrix Σ. The problem of recovering the projector onto an eigenspace of Σ from these observations naturally arises in many applications. Recent technique from [9] helps to study the asymptotic distribution of the distance in the Frobenius norm Pr − Pr 2 between the true projector Pr on the subspace of the rth eigenvalue and its empirical counterpart Pr in terms of the effective rank of Σ. This paper offers a bootstrap procedure for building sharp confidence sets for the true projector Pr from the given data. This procedure does not rely on the asymptotic distribution of Pr − Pr 2 and its moments. It could be applied for small or moderate sample size n and large dimension p. The main result states the validity of the proposed procedure for finite samples with an explicit error bound for the error of bootstrap approximation. This bound involves some new sharp results on Gaussian comparison and Gaussian anti-concentration in high-dimensional spaces. Numeric results confirm a good performance of the method in realistic examples. * Supported by RFBR N 16-31-00005 and Russian President Fellowship for young scientists N 4596.2016.1. † Supported by the Russian Science Foundation grant (project 14-50-00150). MSC 2010 subject classifications: Primary 60K35, 60K35; secondary 60K35

We derive tight non-asymptotic bounds for the Kolmogorov distance between the probabilities of two Gaussian elements to hit a ball in a Hilbert space. The key property of these bounds is that they are dimension-free and depend on the nuclear (Schatten-one) norm of the difference between the covariance operators of the elements and on the norm of the mean shift. The obtained bounds significantly improve the bound based on Pinsker's inequality via the Kullback-Leibler divergence. We also establish an anti-concentration bound for a squared norm of a non-centered Gaussian element in Hilbert space. The paper presents a number of examples motivating our results and applications of the obtained bounds to statistical inference and to high-dimensional CLT.

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