Limit theorems for deviation means of independent and identically distributed random variables

TL;DR

This paper establishes limit theorems such as the law of large numbers, central limit theorem, and large deviation principles for deviation means of i.i.d. random variables, extending classical results to a broader class of estimators.

Contribution

It introduces new, weaker conditions for limit theorems of deviation means, generalizing previous results for quasi-arithmetic and Bajraktarević means.

Findings

Proves strong law of large numbers for deviation means under pairwise independence.

Establishes a central limit theorem for deviation means with weaker assumptions.

Derives large deviation principles for deviation means.

Abstract

We derive a strong law of large numbers, a central limit theorem, a law of the iterated logarithm and a large deviation theorem for so-called deviation means of independent and identically distributed random variables (for the strong law of large numbers, we suppose only pairwise independence instead of (total) independence). The class of deviation means is a special class of M-estimators or more generally extremum estimators, which are well-studied in statistics. The assumptions of our limit theorems for deviation means seem to be new and weaker than the known ones for M-estimators in the literature. Especially, our results on the strong law of large numbers and on the central limit theorem generalize the corresponding ones for quasi-arithmetic means due to de Carvalho (2016) and the ones for Bajraktarevi\'c means due to Barczy and Burai (2022).