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

Motivated by metastability in the zero-range process, we consider i. i. d. \ random variables with values in ₀ and Weibull-like (stretched exponential) law P (Xᵢ =k) = c (- k^), (0, 1). We condition on large values of the sum Sₙ= n + s n^ and prove large deviation principles for the rescaled maximum Mₙ /n^ and for the reversed order statistics. The scale is n^ with = 1/ (2-) ; 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^ = n^2-1, the speed of the large deviation principles. The rate function for Mₙ/n^ is non-convex and solves a recursive equation similar to a Bellman equation.