Abstract
Given a sequence of vectors in a Hilbert space, we propose to use the spectrum of the associated Gram matrix as a tool for extracting statistical information on the sequence. We examine two simple models in some detail: the fractional shift where the sequence is generated by a deterministic unitary dynamics and random normalized vectors in a high-dimensional space chosen at a given density. In both cases, the limiting eigenvalue distribution of the Gram matrix is explicitly found. We relate our results to the notion of growth entropy and recover in the stochastic case the eigenvalue distribution of the Wishart matrices.
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