The number of K-tons in the coupon collector problem

Consider the coupon collector problem where each box of a brand of cereal contains a coupon and there are n different types of coupons. Suppose that the probability of a box containing a coupon of a specific type is $1/n$ , and that we keep buying boxes until we collect at least m coupons of each ty...

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Bibliographic Details
Published in:Journal of applied probability 2023-09, Vol.60 (3), p.723-736
Main Author: Saunders, John C.
Format: Article
Language:English
Subjects:
Asymptotic properties
Boxes
Collectors
Discount coupons
Effectiveness
Expected values
Heuristic
Original Article
Probability
Probability distribution
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Summary:Consider the coupon collector problem where each box of a brand of cereal contains a coupon and there are n different types of coupons. Suppose that the probability of a box containing a coupon of a specific type is $1/n$ , and that we keep buying boxes until we collect at least m coupons of each type. For $k\geq m$ call a certain coupon a k-ton if we see it k times by the time we have seen m copies of all of the coupons. Here we determine the asymptotic distribution of the number of k-tons after we have collected m copies of each coupon for any k in a restricted range, given any fixed m. We also determine the asymptotic joint probability distribution over such values of k, and the total number of coupons collected.
ISSN:0021-9002
1475-6072
DOI:10.1017/jpr.2022.94