Notes on the occupancy problem with infinitely many boxes: general asymptotics and power laws ∗

By Alexander V. Gnedin, Ben B. Hansen, Jim Pitman. 2007

184 citations recorded by the source corpus.

This paper collects facts about the number of occupied boxes in the classical balls-in-boxes occupancy scheme with infinitely many positive frequencies: equivalently, about the number of species represented in sam- ples from populations with infinitely many species. We present moments of this random variable, discuss asymptotic relations among them and with re- lated random variables, and draw connections with regular variation, which appears in various manifestations. AMS 2000 subject classifications: Primary 60F05, 60F15; secondary

Subfields: Condensed Matter Physics, Mathematical Physics, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/ecd107e3c89affd228c3e420db154d0b89952d60

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In Condensed Matter Physics

In Mathematical Physics

In Statistics and Probability

Citations in the corpus

Concentration inequalities in the infinite urn scheme for occupancy counts and the missing mass, with applications (2014, methodology): “Then `1 ∈ rv0 and limt→∞ `1(t)/`(t) =∞ and the following asymptotics hold: 1This is also referred to sometimes as rapid variation, as in Gnedin et al. [2007], but we avoid that terminology as it also has a different connotation in the regular variation literature.”
Beta Processes, Stick-Breaking and Power Laws (2011, background): “The two α > 0 cases clearly adhere to a power law that may be predicted from our results above and the Gnedin et al. [2007] results with C as in Eq.”
Recent progress in coalescent theory (2009, unknown): “For this we will use the following law of large numbers, proved in Proposition 2 of [91].”
Limit theorems for the number of occupied boxes in the Bernoulli sieve (2010, background): “See [5, 10] for survey of recent results on the infinite occupancy.”
LOCAL LIMIT THEOREMS FOR FINITE AND INFINITE URN MODELS (2006, result): “Our results represent an optimal improvement over previous ones for normal approximation.”