Let $x(t)$ and $y(t)$ be independent, mean-zero Gaussian processes, indexed over some general metric space $T$. Is it true that $\Pr(\sup_{t \in T} |x(t) + y(t)| > z) \ge \Pr(\sup_{t \in T} |x(t)| > z)$ for all $z > 0$?
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Don’t the variances add up in the first case? It’s the absolute value of a Gaussian process with mean $0$ but higher variance than just $x$ alone. – Fede Poncio Dec 14 '22 at 00:23