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Self-normalized Cramér type moderate deviations for stationary sequences and applications. (English) Zbl 1445.60027
Summary: Let \((X_i)_{i \geq 1}\) be a stationary sequence. Denote \(m = \lfloor n^\alpha \rfloor, 0 < \alpha < 1,\) and \(k = \lfloor n / m \rfloor\), where \(\lfloor a \rfloor\) stands for the integer part of \(a\). Set \(S_j^\circ = \sum_{i = 1}^m X_{m (j - 1) + i}, 1 \leq j \leq k,\) and \((V_k^\circ)^2 = \sum_{j = 1}^k (S_j^\circ)^2\). We prove a Cramér type moderate deviation expansion for \(\mathbb{P} (\sum_{j = 1}^k S_j^\circ / V_k^\circ \geq x)\) as \(n \to \infty\). Applications to mixing type sequences, contracting Markov chains, expanding maps and confidence intervals are discussed.
MSC:
60F10 Large deviations
60G10 Stationary stochastic processes
60E15 Inequalities; stochastic orderings
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