REFINED CRAMER-TYPE MODERATE DEVIATION THEOREMS FOR GENERAL SELF-NORMALIZED SUMS WITH APPLICATIONS TO DEPENDENT RANDOM VARIABLES AND WINSORIZED MEAN

Let {(Xi, Yi)}(i=1)(n) be a sequence of independent bivariate random vec- tors. In this paper, we establish a refined Cramer-type moderate deviation theorem for the general self-normalized sum Sigma(n)(i=1) X-i/(Sigma(n)(i=1) Y-i(2))(1/2), which unifies and extends the classical Cramer (Actual. Sci. Ind. 736 (1938) 5-23) theorem and the self-normalized Cramer-type moderate deviation theorems by Jing, Shao and Wang (Ann. Probab. 31 (2003) 2167-2215) as well as the further refined version by Wang (J. Theoret. Probab. 24 (2011) 307-329). The advantage of our result is evidenced through successful applications to weakly dependent random variables and self-normalized winsorized mean. Specifically, by applying our new framework on general self-normalized sum, we significantly improve Cramer-type moderate deviation theorems for one-dependent random variables, geometrically beta-mixing random variables and causal processes under geometrical moment contraction. As an additional application, we also derive the Cramer-type moderate deviation theorems for self-normalized winsorized mean.