On the tail behaviour of quantile processes (Q1092511)

For a distribution function F(x), let \(Q(t)=Q_ F(t)=\inf \{x:F(x)\geq t\}\), \(0<t<1\). Let \(X_ 1,X_ 2,...,X_ n\) be independent random variables with common distribution function F(x). Let \(X_{1:n}\leq X_{2:n}\leq...\leq X_{n:n}\) be the order statistics of the \(X_ j\), and let \(Q_ n(t)=X_{k:n}\) if \((k-1)/n<t\leq k/n\). Let \(k_ n\to +\infty\) with n and \(k_ n/n\to 0\) as \(n\to +\infty.\) The author investigates the relation of \(Q_ n(t)\) and Q(t) in the range \(1/(n+1)\leq t\leq k_ n/n\), and characterizes the limit behaviour of some function of \(Q_ n(t)\) and Q(t) in the light of extreme value theory for the minimum.