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

For an integer $n\geq1$, consider a random partition $Π_{n}$ of $\{1,\ldots,n\}$ into $K_{n}$ partition sets with $K_{r,n}$ partition subsets of size $r=1,\ldots,n$, and assume $Π_{n}$ distributed according to the Ewens-Pitman model with parameters $α\in]0,1[$ and $θ>-α$. 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 $p_α=(p_α(1),p_α(2),\ldots)$, where $p_α(r)$ 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_α)\underset{n\rightarrow+\infty}{\overset{\cL}{\longrightarrow}}\mathcal{G}(Γ_α), \end{displaymath} where $\mathcal{G}(Γ_α)$ stands for a centered Gaussian process with covariance matrix $Γ_α=diag(p_α) - p_α p_α^T$. We apply our result to the estimation of the parameter