Asymptotics deviation probabilities of the sum $S_n=X_1+\dots+X_n$ of independent and identically distributed real-valued random variables have been extensively investigated, in particular when $X_1$ is not exponentially integrable. For instance, Nagaev (1969a, 1969b) formulated exact asymptotics results for $\mathbb{P}(S_n>x_n)$ with $x_n\to \infty$ when $X_1$ has a semiexponential distribution. In the same setting, Brosset et al. (2020) derived deviation results at logarithmic scale with shorter proofs relying on classical tools of large-deviation theory and making the rate function at the transition explicit. In this paper we exhibit the same asymptotic behavior for triangular arrays of semiexponentially distributed random variables.

Necessary and sufficient conditions are found for a triangular array (finite or infinite) to be expected values of order statistics. The conditions are a wellknown recurrence relation, and a moment condition.