The extremes of a triangular array of normal random variables

Abstract

Consider a triangular array of stationary normal random variables ${\xi_{n, i}, i \geq 0, n \geq 1)$ such that ${\xi_{n, i}, i \geq 0}$ is a stationary normal sequence for each $n \geq 1$. Let $\rho_{n, j} = \corr (\xi_{n, i}, \xi_{n, i + j})$. We show that if $(1 - \rho_{n,j}) \log n \to \delta_j \epsilon (0, \infty)$ as $n \to \infty$ for some j, then the locations where the extreme values occur cluster, and if $\rho_{n,j}$ tends to 0 fast enough as $j \to \infty$ for fixed n, then ${\xi_{n, i}, i \geq 0}$ satisfies a certain weak dependence condition. Under the two conditions, it is possible to speak about an index which measures the degree of clustering. In practice, this viewpoint can provide a better approximation of the distributions of the maxima of weakly dependent normal random variables than what is directly guided by the asymptotic theory of Berman.