Abstract
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, where sum _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.
Highlights
We study the following classical urn model first considered by Karlin [12]: n ≥ 1 balls are distributed one by one over an infinite number of urns enumerated from 1 to infinity
The ball distributed at step j = 1, 2 . . . , call it jth ball, gets into urn i with probability pi
1, independently of the other balls. Such multinomial occupancy schemes arise in many different applications, in Biology [11], Computer science [13,14] and in many other areas, see, e.g., [10] and the references therein
Summary
We study the following classical urn model first considered by Karlin [12]:n ≥ 1 balls are distributed one by one over an infinite number of urns enumerated from 1 to infinity. The paper extends the results of [6] and [7], where a functional central limit theorem (FCLT) was shown under condition (3) for the vector process
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