Large deviations for the maximum and reversed order statistics of Weibull-like variables

TL;DR

This paper establishes large deviation principles for the maximum and reversed order statistics of Weibull-like variables conditioned on large sums, revealing non-convex rate functions and connections to metastability phenomena.

Contribution

It introduces large deviation results for Weibull-like variables conditioned on large sums, with novel recursive rate functions and analysis of the maximum's behavior.

Findings

Large deviation principles are proven for the maximum and reversed order statistics.

The rate function for the maximum is non-convex and satisfies a recursive Bellman-like equation.

The scale for deviations is $n^g$ with $g=1/(2-a)$, linking heavy-tail and normal approximations.

Abstract

Motivated by metastability in the zero-range process, we consider i.i.d.\ random variables with values in $\N_0$ and Weibull-like (stretched exponential) law $\mathbb P(X_i =k) = c \exp( - k^\alpha)$, $\alpha \in (0,1)$. We condition on large values of the sum $S_n= \mu n + s n^\gamma$ and prove large deviation principles for the rescaled maximum $M_n /n^\gamma$ and for the reversed order statistics. The scale is $n^\gamma$ with $\gamma = 1/(2-\alpha)$; on that scale, the big-jump principle for heavy-tailed variables and a naive normal approximation for moderate deviations yield bounds of the same order $n^{\gamma \alpha} = n^{2\gamma-1}$, the speed of the large deviation principles. The rate function for $M_n/n^\gamma$ is non-convex and solves a recursive equation similar to a Bellman equation.