On the minimum of independent collecting processes via the Stirling numbers of the second kind

Aristides V. Doumas

Statistics & Probability Letters, 2022, vol. 185, issue C

Abstract: We consider the problem where p players aim independently to a complete set of N different types of items (species, viruses) which are uniformly distributed. Let the random variables TN(i),i=1,2,…,p denoting the number of trials needed until all N types are detected (at least once), respectively for each player. This paper studies the impact of the number p in the asymptotics of the expectation, the second moment, and the variance of the random variable MN(p):=⋀i=1pTN(i),N→∞.The main ingredient in the expression of these quantities are sums involving the Stirling numbers of the second kind; for which the asymptotics are explored. At the end of the paper we conjecture on a remarkable identity, regarding alternating binomial sums. These sums have been studied (mainly) by P. Flajolet due to their applications to digital search trees and quadtrees.

Keywords: Species detection; Viruses detection; Coupon collector’s problem; Stirling numbers of the second kind; Digital search trees; Quadtrees (search for similar items in EconPapers)
Date: 2022
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DOI: 10.1016/j.spl.2022.109426

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