Almost sure limit points of independent copies of sample maxima (Q1063929)

Let \(X_ 1,X_ 2,..\). be iid random variables, \(Y_ n=\max (X_ 1,...,X_ n)\), and \(Y_{1,n}\), \(Y_{2,n}\) two independent copies of \(Y_ n\). Almost sure limit sets for the properly normalized vectors \((Y_{1,n},Y_{2,n})\) are derived. Under conditions of \textit{L. de Haan} and \textit{A. Hordijk} [Ann. Math. Stat. 43, 1185-1196 (1972; Zbl 0241.60033)] the limit sets are \(\{\) (x,y):\(1\leq x\leq e^ c\), \(1\leq y\leq e^ c\), \(xy\leq e^ c\}\) and \(\{\) (x,y):\( 0\leq x\leq 1\), \(0\leq y\leq 1\), \(x+y\leq 1\}\), respectively.