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 exp( - k^α), α ∈ (0,1). We condition on large values of the sum Sₙ= μ n + s n^γ and prove large deviation principles for the rescaled maximum Mₙ /n^γ and for the reversed order statistics. The scale is n^γ with γ = 1/(2-α); 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 nγ α = n2γ-1, the speed of the large deviation principles. The rate function for Mₙ/n^γ is non-convex and solves a recursive equation similar to a Bellman equation.