A Gaussian process limit for the self-normalized Ewens-Pitman process

Abstract

For an integer $n\geq1$, consider a random partition $\Pi_{n}$ of $\{1,\ldots,n\}$ into $K_{n}$ partition sets with $K_{r,n}$ partition subsets of size $r=1,\ldots,n$, and assume $\Pi_{n}$ distributed according to the Ewens-Pitman model with parameters $\alpha\in]0,1[$ and $\theta>-\alpha$. Although the large-$n$ asymptotic behaviors of $K_{n}$ and $K_{r,n}$ are well understood in terms of almost sure convergence and Gaussian fluctuations, much less is known about the asymptotic behavior of $P_{r,n}=K_{r,n}/K_n$ and of the self-normalized Ewens-Pitman process $(P_{1,n},P_{2,n},\dots)$. Motivated by the almost sure convergence of $(P_{1,n},P_{2,n},\dots)$ to the Sibuya distribution $p_{\alpha}=(p_{\alpha}(1),p_{\alpha}(2),\ldots)$, where $p_{\alpha}(r)$ is the probability mass at $r=1,2,\ldots$, we establish the $\ell^{2}$ distributional convergence \begin{displaymath} \sqrt{K_{n}}((P_{1,n},\,P_{2,n},\ldots)-p_{\alpha})\underset{n\rightarrow+\infty}{\overset{\cL}{\longrightarrow}}\mathcal{G}(\Gamma_\alpha), \end{displaymath} where $\mathcal{G}(\Gamma_\alpha)$ stands for a centered Gaussian process with covariance matrix $\Gamma_\alpha=diag(p_{\alpha}) - p_{\alpha} p_{\alpha}^T$. We apply our result to the estimation of the parameter