# The impatient collector

## TL;DR

This paper studies the shape of the completion curve in the coupon collector problem under the condition of completing the collection unusually quickly, and applies findings to automata theory.

## Contribution

It introduces the asymptotic shape of the completion curve conditioned on rapid collection, extending classical results and deriving a new formula for automata analysis.

## Key findings

- Characterizes the asymptotic completion curve under fast collection conditions
- Provides a new derivation of Koršunov's formula for automata
- Enhances understanding of collection dynamics under atypical scenarios

## Abstract

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.