Modern Stochastics: Theory and Applications

We consider an extended variant of the classical coupon collector’s problem with an infinite number of collections. An arriving coupon is placed in the rth collection, $r\ge 0$, if r is the smallest index such that the corresponding collection still does not have a coupon of this type. We derive distributional limit theorems for the number of empty spots in different collections at the time when the 0th collection was completed, as well as after some delay. We also obtain the joint limiting distribution for completion times of different collections. All main results are given in an ultimate infinite-dimensional form in the sense of distributional convergence in ${\mathbb{R}^{\infty }}$. The main tool in the proofs is convergence of specially constructed point processes.

The term moderate deviations is often used in the literature to mean a class of large deviation principles that, in some sense, fills the gap between a convergence in probability to zero (governed by a large deviation principle) and a weak convergence to a centered normal distribution. In this paper, some examples of classes of large deviation principles of this kind are presented, but the involved random variables converge weakly to Gumbel, exponential and Laplace distributions.