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Asymptotic behavior of functions, asymptotic series and related topics
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Tail bound for maximum of independent (but not identical) binomial random variables
You can derive some conditions applying the Bernstein's trick. For any positive $t$, we have
$$
P\left( \max_{i\le m}X_i > c_{n,m}\right) = P\left( \exp\{t\max_{i\le m}X_i\} > e^{t c_{n,m}}\right) \l …