A Study of the Suprenewal Process

[摘要] The classical coupon collector’s problem is concerned with the number of purchases in order to have a complete collection, assuming that on each purchase a consumer can obtain a randomly chosen coupon. For most real situations, a consumer may not just get exactly one coupon on each purchase. Motivated by the classical coupon collector’s problem, in this work, we study the so-called suprenewal process. Let {Xi , i ≥ 1} be a sequence of independent and identically distributed random variables, ∑ Sn = n i=1 Xi , n ≥ 1, S0 = 0. For every t ≥ 0, define Qt = inf{n | n ≥ 0, Sn ≥ t}. For the classical coupon collector’s problem, Qt denotes the minimal number of purchases, such that the total number of coupons that the consumer has owned is greater than or equal to t, t ≥ 0. First the process {Qt, t ≥ 0} and the renewal process {Nt, t ≥ 0}, where Nt = sup{n|n ≥ 0, Sn ≤ t}, generated by the same sequence {Xi , i ≥ 1} are compared. Next some fundamental and interesting properties of {Qt, t ≥ 0} are provided. Finally limiting and some other related results are obtained for the process {Qt, t ≥ 0}.

[学科分类] 土木及结构工程学

[关键词] Coupon collector’s problem;geometric distribution;renewal process