On the Wassertein distance for a martingale central limit theorem

We prove an upper bound on the Wassertein distance between normalized martingales and the standard normal random variable, which extends a result of R\"ollin [Statist. Probabil. Lett. 138 (2018) 171-176]. The proof is based on a method of Bolthausen [Ann. Probab. 10 (1982) 672-688]

Sharp moderate and large deviations for sample quantiles

In this article, we discuss the sharp moderate and large deviations between the quantiles of population and the quantiles of samples. Cram\'{e}r type moderate deviations and Bahadur-Rao type large deviations are established with some mild conditions. The results refine the moderate and large deviation principles of Xu and Miao [Filomat 2011; 25(2): 197-206].Comment: 10 page

Hoeffding's inequality for supermartingales

We give an extension of Hoeffding's inequality to the case of supermartingales with differences bounded from above. Our inequality strengthens or extends the inequalities of Freedman, Bernstein, Prohorov, Bennett and Nagaev.Comment: 20 pages, accepted; Stochastic Processes and their Applications (2012), Vol. 122, pages 3545-355