30 research outputs found

    Persistence probabilities in centered, stationary, Gaussian processes in discrete time

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    Lower bounds for persistence probabilities of stationary Gaussian processes in discrete time are obtained under various conditions on the spectral measure of the process. Examples are given to show that the persistence probability can decay faster than exponentially. It is shown that if the spectral measure is not singular, then the exponent in the persistence probability cannot grow faster than quadratically. An example that appears (from numerical evidence) to achieve this lower bound is presented.Comment: 9 pages; To appear in a special volume of the Indian Journal of Pure and Applied Mathematic

    The single ring theorem

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    We study the empirical measure LAnL_{A_n} of the eigenvalues of non-normal square matrices of the form An=UnDnVnA_n=U_nD_nV_n with Un,VnU_n,V_n independent Haar distributed on the unitary group and DnD_n real diagonal. We show that when the empirical measure of the eigenvalues of DnD_n converges, and DnD_n satisfies some technical conditions, LAnL_{A_n} converges towards a rotationally invariant measure on the complex plan whose support is a single ring. In particular, we provide a complete proof of Feinberg-Zee single ring theorem \cite{FZ}. We also consider the case where Un,VnU_n,V_n are independent Haar distributed on the orthogonal group.Comment: Correction of inadequate treatment of neighborhood of z=0 in original submission, various typos corrected following referee's remark
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