Large deviations for a class of recursive algorithms
type: Item
instance of: scholarly article
title: Large deviations for a class of recursive algorithms
zbMATH Open document ID: 0853.60030
DOI: 10.1016/0167-7152(95)00002-X
published in: Statistics \& Probability Letters
publication date: 1996-07-31T00:00:00Z
review text: It is shown that the distributions \(\mu_n\), \(n\in \mathbb{N}\), of the (càdlàg) trajectories \([0, 1]\ni t\mapsto X^{(n)}_{[ nt]} (\omega)\in \mathbb{R}^d\) satisfies a large deviation principle with a good rate function. The random variables \(X_j^{(n)} (\omega)\), \(j=1, \dots, n\), are given by the dynamical system \[ X_0^{(n)} (\omega)= 0, \qquad X^{(n)}_{j+1} (\omega)= X_j^{(n)} (\omega)+ b(X_j^{(n)} (\omega), \xi_n (\omega)), \] where \(\mathbb{R}^d \times \mathbb{R}^d \ni (x, \xi)\mapsto b(x, \xi)\in \mathbb{R}^d\) is bounded, Lipschitz in \(x\), and where \(\{\xi_n \}_{n\in \mathbb{N}}\) are random variables satisfying a certain mixing condition. This condition is implied by various others, e.g. hyper-mixing (cf. Dembo/Zeitouni or Deuschel/Stroock), \(\psi\)-mixing (in the sense of Bryc), or by stationary processes of hyper-exponential \(\alpha\)-mixing rate.
Mathematics Subject Classification ID: 60F10
zbMATH DE Number: 885145
zbMATH Keywords: large deviation principle
zbMATH Keywords: rate function
zbMATH Keywords: stationary processes
zbMATH Keywords: hyper-exponential alpha-mixing rate
author: Tim Zajic
reviewed by: R. L. Schilling
MaRDI profile type: Publication
cites work: On large deviations for uniformly strong mixing sequences
cites work: Large deviations and strong mixing
cites work: Large deviations: From empirical mean and measure to partial sums process
cites work: Large deviations analysis of some recursive algorithms with state dependent noise
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description: scientific article; zbMATH DE number 885145