Suppose $X(t)$ follows a strictly-sense stationary(SSS) Gaussian Process with the mean to be $\mu$ and autovariance $\sigma^2$
How to prove that $\int_{0}^{T}{{X(t)}dt}$ is random variable that follows the Gaussian Distribution $\mathcal{N}(\mu T, \sigma^2 T^2)$
That is to prove that $\int_{0}^{T}{{X(t)}dt} \sim \mathcal{N}(\mu T, \sigma^2 T^2) $
Several theorem that might be helpful
Sum of the independent Gaussian distributed random variable remains to be Gaussian distributed random variable.
Integral of the random process that follows Gaussian process remains to follow Gaussian process.
However sum is a little different from integral. So the theorem 1 may not apply. I feel that integral of strictly-sense stationary(SSS) Gaussian process could be a random variable, but I don't know how to prove that