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

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, ${\sum }_{j=1}^n{a}_j{D}_{n,j}$ for a prescribed sequence of real numbers ${\left(a_j\right)}_{j\in N}$. 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.