Let $X$ be a random variable having a beta distribution $$f(x)=\frac{\Gamma(\alpha)\Gamma(\beta)}{\Gamma(\alpha+\beta)}x^{\alpha-1}(1-x)^{\beta-1}$$with mean $\mu=\frac{\alpha}{\alpha+\beta}$, and assume that $\beta\geq 100$ and $1\leq\alpha\leq\beta/2$. I am trying to find a good upper bound for $\Pr(X\geq c\mu)$ for positive $c$. Is there any work on concentration inequalities of this type? Both the Markov and Chebyshev inequalities were a little too loose for my intended purpose, and I'm hoping to exploit the properties of this particular distribution somehow.
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$\begingroup$ Can we assume $c>1$? Otherwise the probability is not going to zero. The tail Probability you want is an explicit integral. What type of estimate are you hoping for? Is the case where $\alpha$ and $\beta$ are integers of interest to you? $\endgroup$– Yuval PeresCommented Oct 16, 2019 at 3:22
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