Convergence of extremes from normal-skew-normal and minima and maxima from exchangeable trivariate normal distributions

In this article, the extreme-value distribution of the partial maximum for a random sequence of the normal-skew-normal (NSN) distribution, to the Gumbel distribution, is derived. The asymptotic distribution of the normalized maximum and its optimal convergence rate are determined using the optimal choice of normalizing constants. These results are applied to derive the extreme-value distributions of minimum and maximum for the exchangeable trivariate normal random vectors, and lifetimes of parallel and series systems having three dependent log-normal components.