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 \(u > 0\),

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

Proof