Asymptotically balanced functions and stochastic compactness of sample extremes
Let \(X_ n\) be the maximum of the first n terms of a sequence of independent random variables with common distribution function F. The sequence \(\{X_ n\}\) is called stochastically compact if there exist two sequences \(\{a_ n\}\) and \(\{b_ n\}\), \(a_ n>0\), such that every subsequence of \(\{(X_ n-b_ n)/a_ n\}\) contains a further subsequence having proper nondegenerate limit distribution. To obtain conditions for this stochastic compactness the authors introduce a new type of extended regular variation called asymptotic balance. A nondecreasing function \(\Psi\) :(1,\(\infty)\to R\) is called asymptotically balanced if there exists a positive function a(.) such that all partial limits of (\(\Psi\) (tx)-\(\Psi\) (x))/a(t) for \(t\to \infty\) are finite and not identically zero. It is shown that stochastic compactness of \(\{X_ n\}\) is equivalent to asymptotic balance of the inverse function \(\Psi\) of 1/(1-F). Several necessary and sufficient conditions for the stochastic compactness of \(\{X_ n\}\) are given in terms of \(\Psi\) or F itself.
scientific article; zbMATH DE number 3856099