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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). 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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). 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