Let $N_n=\{1,2,...,n\}$. Elements are drawn from the set $N_n$ with
replacement, assuming that each element has probability $1/n$ of being drawn.
We determine the limiting distributions for the waiting time until the given
portion of pairs $jj$, $j\in N_n$, is sampled. Exact distributions of some
related random variables and their characteristics are also obtained.Comment: Published in at http://dx.doi.org/10.3150/07-BEJ114 the Bernoulli
(http://isi.cbs.nl/bernoulli/) by the International Statistical
Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm
We study point processes associated with coupon collector’s problem, that are defined as follows. We draw with replacement from the set of the first n positive integers until all elements are sampled, assuming that all elements have equal probability of being drawn. The point process we are interested in is determined by ordinal numbers of drawing elements that didn’t appear before. The set of real numbers is considered as the state space. We prove that the point process obtained after a suitable linear transformation of the state space converges weakly to the limiting Poisson random measure whose mean measure is determined.
We also consider rates of convergence in certain limit theorems for the problem of collecting pairs.
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