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

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/n0, define the self-normalized trimmed sumT _n= _i=rn ⁿ X _n(i)/{ _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.