30 research outputs found
Persistence probabilities in centered, stationary, Gaussian processes in discrete time
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
We study the empirical measure of the eigenvalues of non-normal
square matrices of the form with independent Haar
distributed on the unitary group and real diagonal. We show that when the
empirical measure of the eigenvalues of converges, and satisfies
some technical conditions, 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 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