On the Extreme Terms of a Sample From the Domain of Attraction of a Stable Law

Let X 1 , X 2 , …, X n be independent random variables having a common distribution in the domain of attraction of a stable law with exponent α < 2. Let { X n 1 , X n 2 , …, X nn } denote the sample { X 1 , X 2 , …, X n } arranged in order of decreasing magnitude:| X n 1| ⩾ | X n 2| ⩾ … ⩾ | X n n| .It is known that the variables∑ 1 kX n jand∑ k + 1 nX n j, when suitably normalised, have a joint limiting distribution ( ( k ) T , T ( k ) ) as n → ∞. We find the characteristic function of T ( k ) and show that for suitable constants c k , ( k ) T − c k has a limiting stable distribution as k →∞. We derive a rate of convergence.