Let $Z_1,\ldots, Z_n$ be standardized Gaussian random variables and denote $\rho_{ij}=\mathbb{E}Z_iZ_j$. Can one give an asymptotically sharp bound for $$\mathbb{P}\,(\max_{1\leq i\leq n}Z_i>x), \quad x>0\,?$$
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1 Answer
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In the case where your process is stationary quite precise bounds are known. For instance this article of Tanguy proves that for a stationary process with covariance function $\phi$ $$ P(|M_{n}-EM_{n}|>t) \leq 6 \exp\left(-ct/\sqrt{max(\phi(n^{\alpha}),1/ \log n)}\right), $$ for some $\alpha$ and $c$ provided $\phi(1) <1$. In the general case in do not know anything.
However you will need some assumptions on your covariance function because if all the $Z_{i}$'s are the same then the classical Gaussian bound $$ P(|M_{n}-EM_{n}|>t) \leq 2 \exp(-ct^{2})$$ is sharp. While in the independent case Tanguy's one is sharp.
