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

Abstract

For an integer $n\ge 1$, consider a random partition ${\hat{I}}_n$ of $\{1,\dots ,n\}$ into $K_n$ partition sets with $K_{r,n}$ partition subsets of size $r=1,\dots ,n$, and assume ${\hat{I}}_n$ distributed according to the Ewens-Pitman model with parameters $\hat{I}\pm \in ]0,1[$ and $\hat{I}¸>-\hat{I}\pm$. 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 $p_{\hat{I}}\pm =(p_{\hat{I}}\pm (1),p_{\hat{I}}\pm (2),\dots )$, where $p_{\hat{I}}\pm (r)$ is the probability mass at $r=1,2,\dots$, we establish the ${\ell }^2$ distributional convergence \begin{displaymath} \sqrt{K_{n}}((P_{1,n},\,P_{2,n},\ldots)-p_α)\underset{n\rightarrow+\infty}{\overset{\cL}{\longrightarrow}}\mathcal{G}(Γ_α), \end{displaymath} where $\mathcal{G}(\hat{I}{“}_{\hat{I}}\pm )$ stands for a centered Gaussian process with covariance matrix $\hat{I}{“}_{\hat{I}}\pm =diag(p_{\hat{I}}\pm )-p_{\hat{I}}\pm p_{\hat{I}}{\pm }^T$. We apply our result to the estimation of the parameter

27 pages