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Register a new userRelative perturbation bounds with applications to empirical covariance operators
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The goal of this paper is to establish relative perturbation bounds, tailored for empirical covariance operators. Our main results are expansions for empirical eigenvalues and spectral projectors, leading to concentration inequalities and limit theorems. Our framework is very general, allowing for a huge variety of stationary, ergodic sequences, requiring only $p > 4$ moments. One of the key ingredients is a specific separation measure for population eigenvalues, which we call the relative rank. Developing a new algebraic approach for relative perturbations, we show that this relative rank gives rise to necessary and sufficient conditions for our concentration inequalities and limit theorems.
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A basic problem in operator theory is to estimate how a small perturbation effects the eigenspaces of a self-adjoint compact operator. In this paper, we prove upper bounds for the subspace distance, taylored for structured random perturbations. As a main example, we consider the empirical covariance operator, and show that a sharp bound can be achieved under a relative gap condition. The proof is based on a novel contraction phenomenon, contrasting previous spectral perturbation approaches.
We explore asymptotically optimal bounds for deviations of Bernoulli convolutions from the Poisson limit in terms of the Shannon relative entropy and the Pearson $chi^2$-distance. The results are based on proper non-uniform estimates for densities. They deal with models of non-homogeneous, non-degenerate Bernoulli distributions.
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