Almost sure limit points of sample extremes of an i.i.d. sequence
Let \((X_ n\), \(n\geq 1)\) be a sequence of independent and identically distributed random variables with a common distribution function F which has a positive derivative for large x and for such x let \[ g(x)=((1- F(x))/f(x))\log \log (1/(1-F(x))). \] We obtain the almost sure limit sets of the random vector consisting of the first two largest values (properly normalized) in a random sample on F in the following two cases: 1) \(\lim_{x\to \infty}g(x)/x=c\) \((0\leq c<\infty),\) 2) F(x) is twice differentiable with \(\lim_{x\to \infty}g'(x)=0\).
scientific article; zbMATH DE number 4042950