Asymptotic behavior of self-normalized trimmed sums: Nonnormal limits II

Abstract

Let {X j} be independent, identically distributed random variables which are symmetric about the origin and have a continuous nondegenerate distributionF. Let {X n(1),...,X n(n)} denote the arrangement of {X 1,...,X n} in decreasing order of magnitude, so that with probability one, |X n(1)|>|X n(2)|>...> |X n(n)|. For initegersr n→∞ such thatr n/n→0, define the self-normalized trimmed sumT n=Σ i=rn n X n(i)/{Σ i=rn n X n 2 (i)}1/2. Hahn and Weiner(6) showed that under a probabilistically meaningful analytic condition generalizing the asymptotic normality criterion forT n, various nonnormal limit laws forT n arise which are represented by means of infinite random series. The analytic condition is now extended and the previous approach is refined to obtain limits which are mixtures of a normal, a Rademacher, and a law represented by a more general random series. Each such limit law actually arises as can be seen from the construction of a single distribution whose correspondingL(T n ) generates all of the law along different subsequences, at least if {r n} grows sufficiency fast. Another example clarifies the limitations of the basic approach.