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High Dimensional Probability VIIMaximal Inequalities for Dependent Random Variables

High Dimensional Probability VII: Maximal Inequalities for Dependent Random Variables [Maximal inequalities play a crucial role in many probabilistic limit theorem; for instance, the law of large numbers, the law of the iterated logarithm, the martingale limit theorem and the central limit theorem. Let X1, X2, … be random variables with partial sums Sk = X1 + ⋯ + Xk. Then a maximal inequality gives conditions ensuring that the maximal partial sum Mn = max1 ≤ i ≤ n Si is of the same order as the last sum Sn. In the literature there exist large number of maximal inequalities if X1, X2, … are independent but much fewer for dependent random variables. In this paper, I shall focus on random variables X1, X2, … having some weak dependence properties; such as positive and negative In-correlation, mixing conditions and weak martingale conditions.] http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png

High Dimensional Probability VIIMaximal Inequalities for Dependent Random Variables

Part of the Progress in Probability Book Series (volume 71)
Editors: Houdré, Christian; Mason, David M.; Reynaud-Bouret, Patricia; Rosiński, Jan
Hoffmann-Jørgensen, Jørgen
High Dimensional Probability VII — Sep 22, 2016
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References (21)

Publisher
Springer International Publishing
Copyright
© Springer International Publishing Switzerland 2016
ISBN
978-3-319-40517-9
Pages
61 –104
DOI
10.1007/978-3-319-40519-3_4
Publisher site
See Chapter on Publisher Site

Abstract

[Maximal inequalities play a crucial role in many probabilistic limit theorem; for instance, the law of large numbers, the law of the iterated logarithm, the martingale limit theorem and the central limit theorem. Let X1, X2, … be random variables with partial sums Sk = X1 + ⋯ + Xk. Then a maximal inequality gives conditions ensuring that the maximal partial sum Mn = max1 ≤ i ≤ n Si is of the same order as the last sum Sn. In the literature there exist large number of maximal inequalities if X1, X2, … are independent but much fewer for dependent random variables. In this paper, I shall focus on random variables X1, X2, … having some weak dependence properties; such as positive and negative In-correlation, mixing conditions and weak martingale conditions.]

Published: Sep 22, 2016

Keywords: Demi-martingales; Integral orderings; Mixing conditions; Negative and positive correlation

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