Concentration in the Generalized Chinese Restaurant Process

The Generalized Chinese Restaurant Process (GCRP) describes a sequence of exchangeable random partitions of the numbers $\{1,\dots ,n\}$. This process is related to the Ewens sampling model in Genetics and to Bayesian nonparametric methods such as topic models. In this paper, we study the GCRP in a regime where the number of parts grows like $n^{\alpha }$ with $\alpha >0$. We prove a non-asymptotic concentration result for the number of parts of size $k=o(n^{\alpha \mathrm{/}(2\alpha +4)}\mathrm{/}(\log n{)}^{1\mathrm{/}(2+\alpha )})$. In particular, we show that these random variables concentrate around $c_k{V}_{\ast }{n}^{\alpha }$ where $V_{\ast }{n}^{\alpha }$ is the asymptotic number of parts and $c_k\approx k^{-(1+\alpha)}$isapositivevaluedependingon$k$. We also obtain finite-$n$ bounds for the total number of parts. Our theorems complement asymptotic statements by Pitman and more recent results on large and moderate deviations by Favaro, Feng and Gao.