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Pré-Publication, Document De Travail Année : 2019

The impatient collector

Résumé

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 Kor\v{s}unov.
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Dates et versions

hal-02164935 , version 1 (25-06-2019)

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Anis Amri, Philippe Chassaing. The impatient collector. 2019. ⟨hal-02164935⟩
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