The impatient collector

In the coupon collector problem with \(n\) items, the collector needs a random number of tries \(T_n\simeq n\ln n\) to complete the collection. Also, after \(nt\) tries, the collector has secured approximately a fraction \(\zeta_\infty(t)=1-e^{-t}\) of the complete collection, so we call \(\zeta_\infty\) the (asymptotic) \emph{completion curve}. In this paper, for \(\nu>0\), we address the asymptotic shape \(\zeta (\nu,.) \) of the completion curve under the condition \(T_n\leq \left( 1+\nu \right) n\), i.e. assuming that the collection is \emph{completed unlikely fast}. As an application to the asymptotic study of complete accessible automata, we provide a new derivation of a formula due to Koršunov.