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.

The joint limiting distribution of maximum of the specific sub-sample and maximum of the complete sample from the first-order auto-regressive process with uniform marginal distributions is obtained in this article. There are considered several examples of partial samples, consisted of non-randomly selected terms of the full sample. It is well known that the uniform AR(1) process is strictly stationary random sequence; it doesn?t satisfy condition of weak dependency, that prohibits clustering of extremes. As a consequence of this property, some interesting conclusions about joint asymptotic distributions are reached. [Projekat Ministarstva nauke Republike Srbije, br. 174012]

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