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

TL;DR

This paper proves a functional central limit theorem for weighted occupancy processes in the Karlin model, with applications to Chinese restaurant process permutations, advancing understanding of their asymptotic behavior.

Contribution

It introduces a new limit theorem for weighted occupancy processes in the Karlin model, extending the analysis to applications involving Chinese restaurant processes with specific parameters.

Findings

Establishes a functional central limit theorem for weighted occupancy processes.

Provides limit theorems for permutations from Chinese restaurant processes.

Offers a framework for analyzing asymptotic properties of complex occupancy models.

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 $(\alpha,\theta)$-seating with $\alpha\in(0,1), \theta>-\alpha$. An example is briefly mentioned here, and full details are provided in an accompanying paper.