We investigate a specific infinite urn scheme first considered by Karlin (1967). We prove functional central limit theorems for the total number of urns with at least k balls for different k.
We study the infinite urn scheme when the balls are sequentially distributed over an infinite number of urns labeled 1,2,... so that the urn j at every draw gets a ball with probability $$p_j$$
p
j
, where $$\sum _j p_j=1$$
∑
j
p
j
=
1
. We prove functional central limit theorems for discrete time and the Poissonized version for the urn occupancies process, for the odd occupancy and for the missing mass processes extending the known non-functional central limit theorems.
We explore a probabilistic model of an artistic text: words of the text are chosen independently of each other in accordance with a discrete probability distribution on an infinite dictionary. The words are enumerated 1, 2, . . ., and the probability of appearing the i'th word is asymptotically a power function. Bahadur proved that in this case the number of different words depends on the length of the text is asymptotically a power function, too. On the other hand, in the applied statistics community, there exist statements supported by empirical observations, the Zipf's and the Heaps' laws. We highlight the links between Bahadur results and Zipf's/Heaps' laws, and introduce and analyse a corresponding statistical test. *
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