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Let ${X(t):t \geq 0}$ be a Gaussian process with mean $0$ and bounded (with probability $1$) sample paths. Borell's Theorem states then that for all $u>0$ we have $$P(\sup_{t \geq 0} X(t)>u) \leq 2 \Psi \left(\frac{u-m}{\sigma_T}\right)$$, where $m$ is the median of $\sup X(t)$, $\sigma_T$ is the supremum of $Var(X(t))$ and $\Psi = 1 - \Phi$ is the tail of a standard normal distribution. I need to show that under the assumptions of this theorem (we can use the theorem as well) $\sup X(t)$ has finite all moments, i.e. $E(\sup X(t))^k$ exists and is finite $\forall k \geq 1$.

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