The Asymptotic Distribution of Trimmed Sums

Abstract

Let $X_{1,n} \leq \cdots \leq X_{n,n}$ be the order statistics of $n$ independent and identically distributed random variables and let $m_n$ and $k_n$ be positive integers such that $m_n \rightarrow \infty, k_n \rightarrow \infty, m_n/n \rightarrow 0$ and $k_n/n \rightarrow 0$ as $n \rightarrow \infty$. We find a necessary and sufficient condition for the existence of normalizing and centering constants $A_n > 0$ and $B_n$ such that the sequence $T_n = A^{-1}_n \big\{\sum_{i=m_n+1}^{n-k_n} X_{i,n} - B_n\big\}$ is stochastically compact and completely describe the possible subsequential limiting distributions. We also give a necessary and sufficient condition for the existence of $A_n$ and $B_n$ such that $T_n$ is asymptotically normal. A variant of Stigler's theorem when $m_n/n \rightarrow \alpha$ and $k_n/n \rightarrow 1 - \beta$, where $0 < \alpha < \beta < 1$, is also obtained as a by-product.