Asymptotic Distributions of Order Statistics and Record Values Under the Marshall-Olkin Parametrization Operation

The Marshall and Olkin (1997) parametrization operation is given by F(alpha) = P(alpha)(F(1)) = F(1)/alpha+(alpha) over barF(1) where F(1) is a given distribution function and alpha > 0, (alpha) over bar = 1 - alpha. In this article, we show that both F(alpha) = P(alpha)(F(1)) and the generating distribution F(1) belong to the same domain of maximal (minimal) (upper record value) (lower record value) attraction. Moreover, it is shown that the weak convergence of any non extreme order statistic (order statistic with variable rank) based on F(1), under given normalizing constants, to a non degenerate limit type implies that the same non extreme order statistic based on the family F(alpha) = P(alpha)(F(1)), alpha not equal 1, under the same normalizing constants, does not converge to any non degenerate distribution and vice versa.