  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.
