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Moderate deviations principle for stationary negatively associated sequences. (English) Zbl 0965.60040

Let \(X_1,X_2,\ldots\) be a stationary sequence of negatively associated real-valued centered r.v.’s. It is assumed that \(\sigma^2 \equiv EX_1^2 + 2\sum_{j=2}^{\infty} EX_1 X_j >0\) and \(\exp\{\delta|X_1|\} < \infty\) for some \(\delta >0\). Suppose also that for some \(N_0 \in \mathbb{N}\) one has \(\inf\{P(AB)/(P(A)P(B))\}=1\) where the \(\inf\) is taken over \(A \in \mathcal{F}_{1,l}\), \(B \in \mathcal{F}_{l+N_0,\infty}\), \(l \in \mathbb{N}\), such that \(P(A)P(B) > 0\) (here \(\mathcal{F}_{a,b} = \sigma\{X_i, a \leq i \leq b\}\), \(1 \leq a \leq b < \infty\), \(\mathcal{F}_{b,\infty}=\sigma\{X_i, i \geq b\}\), \(b \geq 1\)). Then the moderate deviations principle is proved for partial sums \(S_n = \sum_{j=1}^n X_j\), \(n \in \mathbb{N}\), with the rate function \(I(x) = x^2/(2\sigma^2)\), \(x \in \mathbb{R}\).

MSC:

60F10 Large deviations
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