A functional central limit theorem for weighted occupancy processes of the Karlin model

TL;DR

This work analyzes fluctuations of weighted occupancy counts in the Karlin infinite-urn model under regularly varying sampling frequencies $p_j$ with exponent $\alpha\in(0,1)$. It develops a functional central limit theorem for the weighted occupancy process $S_{n,\vec a}$, by employing a Poissonization strategy, constructing a Gaussian limit process $\mathcal Z_{\alpha}$ as a stochastic integral with respect to a Gaussian random measure, and then performing careful de-Poissonization to transfer the result to the original model. The limit process has an explicit covariance structure and, under weight growth constraints, admits a Hölder-continuous version; special choices of weights recover occupancy limits and bi-fractional Brownian motion, highlighting a rich family of Gaussian limits. The methodology enables applications to random permutation matrices arising from Chinese restaurant processes with $(\alpha,\theta)$-seating, linking infinite-urn fluctuations to random matrix theory and spectral statistics via the $\alpha$-diversity parameter.

Abstract

A functional central limit theorem is established for weighted occupancy processes of the Karlin model. The weighted occupancy processes take the form of, with $D_{n,j}$ denoting the number of urns with $j$-balls after the first $n$ samplings, $\sum_{j=1}^na_jD_{n,j}$ for a prescribed sequence of real numbers $(a_j)_{j\in\mathbb N}$. The main applications are limit theorems for random permutations induced by Chinese restaurant processes with $(α,θ)$-seating with $α\in(0,1), θ>-α$. An example is briefly mentioned here, and full details are provided in an accompanying paper.

Theorems & Definitions (31)

• Theorem 1.1
• Remark 1.2
• Remark 1.3
• Remark 1.4
• Theorem 1.5
• Proposition 2.1
• proof
• Remark 2.2
• Theorem 3.1
• Proposition 3.2