Large deviation principles for the Ewens-Pitman sampling model
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Let $M_{l,n}$ be the number of blocks with frequency $l$ in the exchangeable
random partition induced by a sample of size $n$ from the Ewens-Pitman sampling
model. We show that, as $n$ tends to infinity, $n^{-1}M_{l,n}$ satisfies a
large deviation principle and we characterize the corresponding rate function.
A conditional counterpart of this large deviation principle is also presented.
Specifically, given an initial sample of size $n$ from the Ewens-Pitman
sampling model, we consider an additional sample of size $m$. For any fixed $n$
and as $m$ tends to infinity, we establish a large deviation principle for the
conditional number of blocks with frequency $l$ in the enlarged sample, given
the initial sample. Interestingly, the conditional and unconditional large
deviation principles coincide, namely there is no long lasting impact of the
given initial sample. Potential applications of our results are discussed in
the context of Bayesian nonparametric inference for discovery probabilities.