Borel-TIS inequality

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Theorem

Let $T$ be a topological space.

Let $\sequence {f_t}_{t \mathop \in T}$ be a centred (i.e. mean zero) Gaussian process on $T$, such that:

$\norm f_T := \sup_{t \mathop \in T} \size {f_t}$

is almost surely finite.

Let:

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


Then $\map {\operatorname {E} } {\norm f_T}$ and $\sigma_T$ are both finite, and, for each $u > 0$:

$\map {\operatorname {P} } {\norm f_T > \map {\operatorname {E} } {\norm f_T} + u} \le \map \exp {\dfrac {- u^2} {2 \sigma_T^2} }$

Proof