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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}. 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