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Moderate deviations for iterates of expanding maps. (English) Zbl 0935.60019

Kabanov, Yu. M. (ed.) et al., Statistics and control of stochastic processes. The Liptser Festschrift. Papers from the Steklov seminar held in Moscow, Russia, 1995-1996. Singapore: World Scientific. 1-11 (1997).
Let \(\{X_i\}\) denote a realization of a centered stationary process with values in the unit ball of \(\mathbb{R}^d\). Let \(S_n=\sum^n_{i=1}X_i\). Suppose that the normalized empirical means \(\widehat{S_n}=S_n/\sqrt n\) converge in distribution to a Gaussian law of zero mean and covariance matrix \(V\) and define \(I(x)=\sup_{\lambda\in \mathbb{R}^d}\{\langle \lambda,x \rangle- {1\over 2} \langle \lambda,V \lambda\rangle\}\). If there exist \(q>0\) and \(c_0\), \(l_0< \infty\), such that for any \(l\geq l_0\) and \(l\)-separated intervals \(\overline I_i\) there exist independent random variables \(\{\widetilde{S_i}\}\), such that for every \(\theta\in (0,q)\) and \(m<\infty\), \(E(\exp(\theta \sum^m_{i=1} |\sum_{j \in I_i}X_j- \widetilde{S_i} |)) \leq e^{\theta c_0m}\), then \(\widehat{S_n}\) satisfies the moderate deviation principle with rate function \(I(\cdot)\) and \(n^{-1}S_n\to 0\) with exponential tails.
For the entire collection see [Zbl 0912.00039].

MSC:

60F05 Central limit and other weak theorems
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