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Speed and concentration of the covering time for structured coupon collectors

Part of: Designs and configurations Stochastic processes Graph theory Combinatorial probability

Published online by Cambridge University Press:  15 July 2020

Victor Falgas-Ravry ,
Joel Larsson Open the ORCID record for Joel Larsson [Opens in a new window]  and
Klas Markström
Victor Falgas-Ravry*
Affiliation:
Umeå Universitet
Joel Larsson*
Affiliation:
Warwick University
Klas Markström*
Affiliation:
Umeå Universitet
*
*Postal address: Department of Mathematics and Mathematical Statistics, Umeå Universitet. Email: [email protected]
**Postal address: Mathematics Institute, Warwick University. Email: [email protected]
***Postal address: Department of Mathematics and Mathematical Statistics, Umeå Universitet. Email: klas.markströ[email protected]
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Abstract

Let V be an n-set, and let X be a random variable taking values in the power-set of V. Suppose we are given a sequence of random coupons $X_1, X_2, \ldots $ , where the $X_i$ are independent random variables with distribution given by X. The covering time T is the smallest integer $t\geq 0$ such that $\bigcup_{i=1}^t X_i=V$ . The distribution of T is important in many applications in combinatorial probability, and has been extensively studied. However the literature has focused almost exclusively on the case where X is assumed to be symmetric and/or uniform in some way.

In this paper we study the covering time for much more general random variables X; we give general criteria for T being sharply concentrated around its mean, precise tools to estimate that mean, as well as examples where T fails to be concentrated and when structural properties in the distribution of X allow for a very different behaviour of T relative to the symmetric/uniform case.

Keywords

Coupon collectorconcentration inequalitiescombinatorial probability

MSC classification

Primary: 60C05: Combinatorial probability
Secondary: 05B40: Packing and covering 05C80: Random graphs 60G99: None of the above, but in this section
Type
Original Article
Information
Advances in Applied Probability , Volume 52 , Issue 2 , June 2020 , pp. 433 - 462
Copyright
© Applied Probability Trust 2020

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