Almost sure central limit theorem for a nonstationary Gaussian sequence.

Zang, Qing-Pei

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Almost sure convergence for the maximum and the sum of nonstationary Gaussian sequences. (Almost sure convergence for the maximum and the sum of nonstationary guassian sequences.)

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In this paper we focus on the problem of estimating a bounded density using a finite combination of densities from a given class. We consider the Maximum Likelihood Estimator (MLE) and the greedy procedure described by Li and Barron (1999) under the additional assumption of boundedness of densities. We prove an $O (\frac{1}{\sqrt{n}})$ bound on the estimation error which does not depend on the number of densities in the estimated combination. Under the boundedness assumption, this improves the bound of Li...

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The large deviation principle for certain series

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We study the large deviation principle for stochastic processes of the form ${\sum_{k = 1}^{\infty} x_{k} (t) ξ_{k} : t \in T}$ , where ${ξ_{k}}_{k = 1}^{\infty}$ is a sequence of i.i.d.r.v.’s with mean zero and $x_{k} (t) \in ℝ$ . We present necessary and sufficient conditions for the large deviation principle for these stochastic processes in several situations. Our approach is based in showing the large deviation principle of the finite dimensional distributions and an exponential asymptotic equicontinuity condition. In order to get the exponential asymptotic equicontinuity...