Consistency of the $\alpha$-trimming of a probability. Applications to central regions

TL;DR

This paper proves the convergence of alpha-trimmings of empirical probabilities to the population alpha-trimming and applies this to analyze the asymptotic behavior of central regions in probability measures.

Contribution

It establishes the convergence of alpha-trimmings in the weak topology and applies this to the asymptotic analysis of central regions.

Findings

Alpha-trimmings of empirical probabilities converge to the population alpha-trimming.

Convergence occurs in the Painlevé–Kuratowski sense.

Results inform the asymptotic behavior of probability-based central regions.

Abstract

The sequence of $\alpha$-trimmings of empirical probabilities is shown to converge, in the Painlev\'{e}--Kuratowski sense, on the class of probability measures endowed with the weak topology, to the $\alpha$-trimming of the population probability. Such a result is applied to the study of the asymptotic behaviour of central regions based on the trimming of a probability.