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

Matyas Barczy, Zsolt Páles

Published 2021-12-09, updated 2022-12-08Version 2

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).