Asymptotic results for sums and extremes

TL;DR

This paper establishes noncentral moderate deviation principles for sums and maxima of i.i.d. random variables, including unbounded cases and sums of partial minima, filling a gap between convergence types.

Contribution

It introduces new moderate deviation results for sums and maxima of i.i.d. variables, including non-Gaussian limits and unbounded cases, extending existing theory.

Findings

Noncentral moderate deviation for sums and maxima of bounded i.i.d. variables.

Moderate deviation results for unbounded maxima with normalization.

Moderate deviations for sums of partial minima of exponential variables.

Abstract

The term moderate deviations is often used in the literature to mean a class of large deviation principles that, in some sense, fills the gap between a convergence in probability of some random variables to a constant and a weak convergence to a centered Gaussian distribution (when such random variables are properly centered and rescaled). We talk about noncentral moderate deviations when the weak convergence is towards a non-Gaussian distribution. In this paper, we prove a noncentral moderate deviation result for the bivariate sequence of sums and maxima of i.i.d. random variables bounded from above. We also prove a result where the random variables are not bounded from above, and the maxima are suitably normalized. Finally, we prove a moderate deviation result for sums of partial minima of i.i.d. exponential random variables.