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Dependent noise for stochastic algorithms

100%

Doukhan P. , Brandierie O.

Probability and Mathematical Statistics

2004

tom Vol. 24, Fasc. 2

381--399

We introduce different ways of being dependent for the input noise of stochastic algorithms. We are aimed to prove that such innovations allow to use the ODE (ordinary differential equation) method. Illustrations to the linear regression frame and to the law of large numbers for triangular arrays of weighted dependent random variables are also given.

An invariance principle for weakly dependent stationary general models

100%

Doukhan P. , Wintenberger O.

Probability and Mathematical Statistics

2007

tom Vol. 27, Fasc. 1

45--73

The aim of this paper is to refine a weak invariance principle for stationary sequences given by Doukhan and Louhichi [10]. Since our conditions are not causal, our assumptions need to be stronger than the mixing and causal 0-weak dependence assumptions used in Dedecker and Doukhan [5]. Here, if moments of order greater than 2 exist, a weak invariance principle and convergence rates in the CLT are obtained; Doukhan and Louhichi [10] assumed the existence of moments with order greater than 4. Besides the η and к-weak dependence conditions used previously, we introduce a weaker one, λ, which fits the Bernoulli shifts with dependent inputs.