Suppose $X$ is a random variable taking values in $[0,1]$ and following the Beta distribution $B(\alpha, (N-1)\alpha)$ with $\alpha > 1$ and $N$ a positive integer. I am looking for the slowest decaying $f:\mathbb{R}\to\mathbb{R}$ such that $\mathbb{P}(X\leq f(N)) = o(1/N)$. Based on my simulations, I suspect $f^*(N)= N^{-\frac{\alpha + 1}{\alpha}}$ may be the limiting rate. Is there any way to formalize this, given that the CDF of the Beta distribution is not really tractable ?
2026-03-28 02:41:13.1774665673
Decaying Beta Random Variable
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