Large deviation principles for the Ewens-Pitman sampling model

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. In this paper 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 observed sample of size $n$ from the Ewens-Pitman sampling model, we consider an additional unobserved sample of size $m$ thus giving rise to an enlarged sample of size $n+m$. Then, 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 this conditional large deviation principle coincides with the large deviation principle for $M_{l,n}$, namely there is no long lasting impact of the given initial sample to the large deviations. Potential applications of our conditional large deviation principle are thoroughly discussed in the context of Bayesian nonparametric inference for species sampling problems.