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1. Cram\'{e}r moderate deviation expansion for martingales with one-sided Sakhanenko's condition and its applications

Author

Fan, Xiequan, Grama, Ion, and Liu, Quansheng

Subjects
Mathematics - Probability, 60G42, 60F10, 60E15, 60F05
Abstract

We give a Cram\'{e}r moderate deviation expansion for martingales with differences having finite conditional moments of order $2+\rho, \rho \in (0,1],$ and finite one-sided conditional exponential moments. The upper bound of the range of validity and the remainder of our expansion are both optimal. Consequently, it leads to a "half-side" moderate deviation principle for martingales. It is worth mentioning that our result is new even for independent random variables. Moreover, applications to quantile coupling inequality, $\beta$-mixing and $\psi$-mixing sequences are discussed., Comment: 24 pages

Published
2019
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2. Cram\'{e}r moderate deviation expansion for martingales with one-sided Sakhanenko's condition and its applications

Author

Ion Grama, Quansheng Liu, Xiequan Fan, Tianjin University (TJU), Laboratoire de Mathématiques de Bretagne Atlantique (LMBA), Université de Bretagne Sud (UBS)-Université de Brest (UBO)-Centre National de la Recherche Scientifique (CNRS), and Center for Applied Mathematics, Tianjin University

Subjects
Statistics and Probability, Pure mathematics, 60G42, 60F10, 60E15, 60F05, General Mathematics, 010102 general mathematics, Order (ring theory), Coupling (probability), 01 natural sciences, Upper and lower bounds, Exponential function, [MATH.MATH-PR]Mathematics [math]/Probability [math.PR], 010104 statistics & probability, Range (statistics), Moderate deviations, 0101 mathematics, Statistics, Probability and Uncertainty, Remainder, Mathematics - Probability, ComputingMilieux_MISCELLANEOUS, Quantile, Mathematics
Abstract

We give a Cram\'{e}r moderate deviation expansion for martingales with differences having finite conditional moments of order $2+\rho, \rho \in (0,1],$ and finite one-sided conditional exponential moments. The upper bound of the range of validity and the remainder of our expansion are both optimal. Consequently, it leads to a "half-side" moderate deviation principle for martingales. It is worth mentioning that our result is new even for independent random variables. Moreover, applications to quantile coupling inequality, $\beta$-mixing and $\psi$-mixing sequences are discussed., Comment: 24 pages

Published
2019

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