Refined Cramér-type moderate deviation theorems for general self-normalized sums with applications to dependent random variables and winsorized mean

Let {(Xi,Yi)}i=1n be a sequence of independent bivariate random vectors. In this paper, we establish refined Cramér-type moderate deviation theorem for the general self-normalized sum ∑ i=1nXi/(∑i=1nYi2)1/2, which unifies and extends classical Cramér (Actual. Sci. Ind. 736 (1938) 5–23) theorems by Jing, Shao Wang (Ann. Probab. 31 (2003) 2167–2215) as well further version (J. Theoret. 24 (2011) 307–329). The advantage our result is evidenced through successful applications to weakly dependent variables winsorized mean. Specifically, applying new framework on sum, significantly improve one-dependent variables, geometrically β-mixing causal processes under geometrical moment contraction. As an additional application, also derive