Some examples of non-central moderate deviations for sequences of real random variables

TL;DR

This paper explores classes of large deviation principles for sequences of real random variables that converge weakly to Gumbel, exponential, and Laplace distributions, filling the gap between convergence in probability and weak convergence.

Contribution

It provides new examples of moderate deviation principles involving non-normal limit distributions such as Gumbel, exponential, and Laplace.

Findings

Examples of large deviation principles with Gumbel, exponential, and Laplace limits

Illustrates the gap between convergence in probability and weak convergence

Expands understanding of moderate deviations beyond normal limits

Abstract

The term \emph{moderate deviations} is often used in the literature to mean a class of large deviation principles that, in some sense, fill the gap between a convergence in probability to zero (governed by a large deviation principle) and a weak convergence to a centered Normal distribution. In this paper we present some examples of classes of large deviation principles of this kind, but the involved random variables converge weakly to Gumbel, exponential and Laplace distributions.