Cramer Type Large Deviations for Linear Combinations of Order Statistics

Abstract

Let $T_n = n^{-1} \sum^n_{i=1} c_{in}X_{in}$ be a linear combination of order statistics and put $T^\ast_n = (T_n - E(T_n))/ \sqrt{\operatorname{Var}(T_n)}$. Sufficient conditions on the $c_{in}$ and on the moments of the underlying distribution are established under which the ratio $P(T^\ast_n > x)/(1 - \Phi(x))$ tends to 1, either uniformly in the range $-A \leq x \leq c \sqrt{\ln n}(A \geq 0, c > 0)$ (moderate deviation theorem) or uniformly in the range $-A \leq x \leq o(n^\alpha)(A \geq 0)$ (Cramer type large deviation theorem). The proof relies on Helmers' approximation method and on the corresponding results for U-statistics.