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Is there a theorem that says if every realization $X(t)$ of a random process $X_{\omega}(t)$ satisfies some property, then the expectation $\mathbb{E}X(t)$ also satisfies the same property?

What about the case where $X$ is a stationary process?

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In general, no. Say with probability $1/2$, $X(t) = 1$ and with probability $1/2$, $X(t) = 0$. Then every realization of the process verifies the property "$X$ takes integer values", while $\mathbb{E}X(t) = 1/2$ doesn't verify that property.

If you specify what a "property" is, you can get such results. If your property says "$X(t)$ takes values in a convex set $A \subset \mathbb{R}^n$" then your assertion is true (see this question).

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