Central limit theorems for an Indian buffet model with random weights

TL;DR

This paper extends the Indian buffet process by incorporating random weights for customers and establishes stable limit theorems for the number of dishes tried and the average dishes per customer, including the standard case.

Contribution

It introduces a generalized Indian buffet process with random weights and proves new stable limit theorems for key quantities, enhancing understanding of its asymptotic behavior.

Findings

Asymptotic distributions of $L_n$ and $ar{K}_n$ are derived.

Limit theorems are stable, not just in distribution.

Results include the standard Indian buffet process as a special case.

Abstract

The three-parameter Indian buffet process is generalized. The possibly different role played by customers is taken into account by suitable (random) weights. Various limit theorems are also proved for such generalized Indian buffet process. Let $L_n$ be the number of dishes experimented by the first $n$ customers, and let $\overline{K}_n=(1/n)\sum_{i=1}^nK_i$ where $K_i$ is the number of dishes tried by customer $i$. The asymptotic distributions of $L_n$ and $\overline{K}_n$, suitably centered and scaled, are obtained. The convergence turns out to be stable (and not only in distribution). As a particular case, the results apply to the standard (i.e., nongeneralized) Indian buffet process.