Lithuanian Mathematical Journal

, Volume 58, Issue 1, pp 54–68 | Cite as

Moderate deviation principle form-dependent random variables

Moderate deviation principle for m-dependent random variables*

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Abstract

Let {X n }n ≥ 1 be a sequence of strictly stationary m-dependent random variableswith EX1 = 0 and \( \mathrm{E}{X}_1^2<\infty \), and let (b n ) be an increasing sequence of positive numbers such that b n  ↑ ∞ and \( {b}_n/\sqrt{n}\downarrow 0\kern0.5em \mathrm{as}\kern0.5em n\to \infty \). We establish a moderate deviation principle of \( {\left({b}_n\sqrt{n}\right)}^{-1}{\sum}_{i=1}^n{X}_i \) under the condition
$$ \underset{n\to \infty }{\lim \sup}\frac{1}{b_n^2}\log \left[n\mathbf{P}\left(\left|{X}_1\right|>{b}_n\sqrt{n}\right)\right]=-\infty, $$

which is weaker than the classical exponential integrability condition. The results in the present paper weaken the assumptions of Chen [5] and extend partially the results of Eichelsbacher and Löwe [10].

Keywords

moderated deviation principle m-dependent random variables 

MSC

60F10 

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© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.College of Mathematics and Information ScienceHenan Normal UniversityXinxiang ShiChina

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