Large deviations for a class of recursive algorithms
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.
scientific article; zbMATH DE number 885145