# Borel-TIS inequality

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## Statement

Let $T$ be a topological space, and let $\{ f_t \}_{t \in T}$ be a centred (i.e. mean zero) Gaussian process on $T$, with

$\| f \|_T := \sup_{t \in T} | f_t |$

almost surely finite, and let

$\sigma_T^2 := \sup_{t \in T} \operatorname{E}| f_t |^2.$

Then $\operatorname{E}(\| f \|_T)$ and $\sigma_T$ are both finite, and, for each $u > 0$,

$\operatorname{P} \big( \| f \|_T > \operatorname{E}(\| f \|_T) + u \big) \leq \exp\left( \frac{- u^2}{2\sigma_T^2} \right).$