“Asymptotic Independence of Certain Statistics Connected With the Extreme Order Statistics in a Bivariate Distribution”, 1967-06 (; backlinks; similar):
The exact distribution of extremes in a sample and its limiting forms are well known in the univariate case. The limiting form for the largest observation in a sample was derived by 1928 as early as 1927 by a functional equation, and that for the smallest was studied by 1952.
Though the joint distribution of two extremes has not been fully studied yet1960 gave a necessary and sufficient condition for the asymptotic independence of two largest extremes in a bivariate distribution.
In this paper a necessary and sufficient condition for the asymptotic independence of two smallest observations in a bivariate sample has been derived, and the result has been used to find the condition for the asymptotic independence of any pair of extreme order statistics, one in each component of the bivariate sample. This result is further extended to find the condition for asymptotic independence of the pair of distances between two order statistics, arising from each component.