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

For an integer $n\ge 1$, consider a random partition ${\mathrm{\Pi }}_n$ of $\{1,\dots ,n\}$ into $K_n$ partition sets with $K_{r,n}$ partition subsets of size $r=1,\dots ,n$, and assume ${\mathrm{\Pi }}_n$ 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 $P_{r,n}=K_{r,n}\mathrm{/}{K}_n$ 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 $p_{\alpha }=(p_{\alpha }(1),p_{\alpha }(2),\dots )$, where $p_{\alpha }(r)$ is the probability mass at $r=1,2,\dots$, we establish the ${\mathrm{\ell }}^2$ 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 $\mathcal{G}({\mathrm{\Gamma }}_{\alpha })$ stands for a centered Gaussian process with covariance matrix ${\mathrm{\Gamma }}_{\alpha }=diag(p_{\alpha })-p_{\alpha }{p}_{\alpha }^T$. We apply our result to the estimation of the parameter