Let $p_1,\ldots,p_k$ be the multinomial parameters sampled from a Dirichlet distribution. Assume $\bar{p_1},\ldots,\bar{p}_k$ be the mean of Dirichlet distribution. Then, $$Pr( \sum_{i}|p_i-\bar{p}_i|\le \epsilon) \ge \prod_{i} Pr (|p_i-\bar{p}_i|\le \epsilon/k)$$

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    $\begingroup$ is this under self-study? $\endgroup$
    – Xi'an
    Feb 4 '15 at 18:43
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    $\begingroup$ @Xi'an Yes, this is. Do you know any reference that talks about this? Does this inequality have any name? $\endgroup$ Feb 4 '15 at 20:06

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