Limit laws of modulus trimmed sums

Abstract

Let $X,X_1,X_2,\ldots$ be a sequence of independent and identically distributed random variables. Let $ ^{(1)}X_n,\ldots,{^{(n)}X}_n$ be an arrangement of $X_1$, $X_2,\ldots,X_n $ in decreasing order of magnitude, and set ${^{(r_n)}S}_n= {}^{(r_{n}+1)}X_n+\cdots + {^{(n)}X}_{n}$. This is known as the modulus trimmed sum. We obtain a complete characterization of the class of limit laws of the normalized modulus trimmed sum when the underlying distribution is symmetric and $ r_n \to \infty$, $r_nn^{-1}\to 0$.