On the Asymptotic Behavior of the Maximum and Record Values of Multivariate Data Using the R-ordering principle

Abstract

By using a sup-norm, sufficient conditions for the convergence of multivariate extremes and the potential limit types were fully identified by Barakat et al. (2020a). In this paper, we prove an intriguing result that by using the sup-norm, the weak convergence of multivariate extremes to the Fréchet type implies the convergence of those multivariate extremes in an arbitrary D-normto the same type-limit by using the same normalizing constants. As a result of this finding, the weak convergence to the Fréchet type takes place by employing any logistic norm. Moreover, the two other possible limit types (max-Weibull and Gumbel types) are discussed. Similar findings are also demonstrated for multivariate record values. Finally, we demonstrate in a real-world scenario how to model multivariate extreme data sets utilizing the R-ordering principle and different norms.

Keywords

• weak convergence
• multivariate extremes
• reduced ordering principle
• sup-norm
• logistic-norm
• d-norm