Abstract
We consider the self-normalized sums $T_{n}=\sum_{i=1}^{n}X_{i}Y_{i}/\sum_{i=1}^{n}Y_{i}$, where ${Y_{i} : i\geq 1}$ are non-negative i.i.d. random variables, and ${X_{i} : i\geq 1} $ are i.i.d. random variables, independent of ${Y_{i} : i \geq 1}$. The main result of the paper is that each subsequential limit law of T_n$ is continuous for any non-degenerate $X_1$ with finite expectation, if and only if $Y_1$ is in the centered Feller class.
Highlights
Let {Y, Yi : i ≥ 1} denote a sequence of i.i.d. random variables, where Y is nonnegative and non-degenerate with cumulative distribution function [cdf] G
In statistics Tn has uses as a version of the weighted bootstrap, where typically more assumptions are imposed on X and Y
We shall see that Tn is an interesting random variable, which is worthy of study in its own right
Summary
Let {Y, Yi : i ≥ 1} denote a sequence of i.i.d. random variables, where Y is nonnegative and non-degenerate with cumulative distribution function [cdf] G. At the end of his paper Breiman conjectured that Tn converges in distribution to a non-degenerate law for some X ∈ X if and only if Y ∈ D (β) , with 0 ≤ β < 1. Mason and Zinn [18] partially verified his conjecture They established the following: Whenever X is non-degenerate and satisfies E|X|p < ∞ for some p > 2, Tn converges in distribution to a non-degenerate random variable if and only if (1.1) holds. Y is in the centered Feller class, if Y is in the Feller class and one can choose bn = 0, for all n ≥ 1 This we shall denote as Y ∈ Fc. In this paper the norming sequence {an} is always assumed to be strictly positive and to tend to infinity. We shall soon see that the innocuous looking sequence of stochastic variables {Tn} displays quite a variety of subsequential distributional limit behavior
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