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

Abstract

Let ${\{({\mathit{X}}_{\mathit{i}},{\mathit{Y}}_{\mathit{i}})\}}_{\mathit{i}=1}^{\mathit{n}}$ be a sequence of independent bivariate random vectors. In this paper, we establish a refined Cramér-type moderate deviation theorem for the general self-normalized sum ${\sum }_{\mathit{i}=1}^{\mathit{n}}{\mathit{X}}_{\mathit{i}}/{({\sum }_{\mathit{i}=1}^{\mathit{n}}{\mathit{Y}}_{\mathit{i}}^2)}^{1/2}$, which unifies and extends the classical Cramér (Actual. Sci. Ind. 736 (1938) 5–23) theorem and the self-normalized Cramér-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 Cramér-type moderate deviation theorems for one-dependent random variables, geometrically β-mixing random variables and causal processes under geometrical moment contraction. As an additional application, we also derive the Cramér-type moderate deviation theorems for self-normalized winsorized mean.