# What happens with the beta-binomial distribution, when n approaches infinity?

Short question: What happens to the beta-binomial distribution, when n increases to infinity? Is there a count distribution arising like it's for the classical binomial distribution?

When both $\alpha$ and $\beta$ are very large, the additional balls introduced do not appreciably change the chances of red or black. This is just a binomial experiment (with fixed probabilities $\alpha / (\alpha + \beta)$ of red and $\beta/ (\alpha + \beta)$ of black).
• I think $\alpha, \beta$ are fixed and he's sending $n\rightarrow\infty$ – JMS Jun 14 '11 at 17:02
• @JMS If that's the case, you recover the underlying Beta distribution if you divide by $n$. – whuber Jun 14 '11 at 18:12
• Right; without additional scaling the beta-binomial won't converge to a proper probability distribution as $n\rightarrow\infty$ if $\alpha, \beta$ are fixed (I think). From his phrasing ("count distribution") I think the OP may be looking for the analogue of a binomial approaching a Poisson as $n\rightarrow \infty$ if $p\rightarrow 0$ and $np$ is constant. Probably there is a similar result if you send $\alpha/(\alpha+\beta)$ to zero at a proper rate or something, but I don't know it. – JMS Jun 14 '11 at 20:29