Journal of Applied Mathematics & Bioinformatics

Notes on the estimation of the asymptotics of the moments for the m collector’s problem

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  • Abstract
    The general collector’s problem describes a process in which N distinct coupons are placed in an urn and randomly selected one at a time (with replacement) until at least m of all N existing different types of coupons have been selected. Let Tm(N) the random variable denoting the number of trials needed for this goal. We briefly present the leading asymptotics of the (rising) moments of Tm(N) as N → ∞ for large classes of coupon probabilities. It is proved that the expectation of Tm(N) becomes minimum when the coupons are uniformly distributed. Moreover, a theorem on the asymptotic estimates of the rising moments of Tm(N) by comparison with known sequences of coupon probabilities is proved.

    Mathematics Subject Classification: 78M05, 60F99, 41A60 
    Keywords: Urn problems, coupon collector’s problem, double Dixie cup problem, rising moments, Zipf law, Schur functions, Schur - Ostrowski criterion.