On the variance of the number of occupied boxes

Abstract

We consider the occupancy problem where balls are thrown independently at infinitely many boxes with fixed positive frequencies. It is well known that the random number of boxes occupied by the first n balls is asymptotically normal if its variance Vn tends to infinity. In this work, we mainly focus on the opposite case where Vn is bounded, and derive a simple necessary and sufficient condition for convergence of Vn to a finite limit, thus settling a long-standing question raised by Karlin in the seminal paper of 1967. One striking consequence of our result is that the possible limit may only be a positive integer number. Some new conditions for other types of behavior of the variance, like boundedness or convergence to infinity, are also obtained. The proofs are based on the poissonization techniques.

Small counts in the infinite occupancy scheme

“…In contrast, for a sequence of geometric frequencies p j = cq j (0 < q < 1), there is no way to scale the X n,r 's to obtain a nontrivial limit distribution [10], and the moments of K n have oscillatory asymptotics. In a more general setting such that the p j 's have exponential decay, the oscillatory behaviour of Var[K n ] is typical [3]. The spectrum of interesting possibilities is, however, much wider: for instance, frequencies p j ∼ ce −j variation condition holds if Var X n,r → ∞ for all r and if all the correlations {Corr (X n,r , X n,s ), r, s ≥ 1} converge.…”

On descents after maximal values in samples of discrete random variables

“…random variables behaves in a similar way: it is a sum of a regularly growing function f (n) which grows roughly as 15 const · log n, a periodic part of small amplitude depending on f (n), and a vanishing remainder term. It is not known whether 16 this type of behaviour is always the case. Actually, this behaviour is even hard to formalize, unless you examine the proof 17 and learn that the main and periodic parts come from different (correspondingly real and imaginary) poles of a complex 18 function, in course of an asymptotic evaluation of certain complex integral.…”

“…It is not known whether 16 this type of behaviour is always the case. Actually, this behaviour is even hard to formalize, unless you examine the proof 17 and learn that the main and periodic parts come from different (correspondingly real and imaginary) poles of a complex 18 function, in course of an asymptotic evaluation of certain complex integral. Nevertheless, the appearance of the periodic 19 component is quite common in problems concerning big samples of discrete random variables, see e.g.…”

“…On the other side, in general we allow Z h = 0 with positive probability. 16 We obtained expressions (6) and (7) assuming that p h > 0. However if p h = 0 then r h = r h−1 , consequently the right- 17 hand sides of both (6) and (7) equals 0 whatever one means by z h,j andz h,j which are undefined in this case.…”

Gaps in Discrete Random Samples

“…The classical models have a finite number of urns and have been extensively studied, but the infinite case has already been considered in [Ka67]. These models have received some attention recently; see [BGY08] and [HJ08], for example. The latter gives a local limit theorem for the number of occupied urns, which is the cardinality of A n in our notation.…”