We can define the integration of Brownian motion $W_t$ wrt t as $$Z_t=\int_0^t W_sds.$$
We have know that $Z_t\sim N(0, t^3/3)$, but I am still not so sure that $Z_t$ is a Gaussian process or not. I don't have clue to follow by definition of a Gaussian process.
Indeed, we can define the integral in Riemann sense: for $0\le i\le n$, let $t_i=\frac {t} ni$, then \begin{equation*} Z_t=\lim_{n\rightarrow \infty}\sum_{i=0}^{n-1}\frac {t} nW_{\frac {t} ni}. \end{equation*} Since $W$ is a Gaussian process, the linear combination $\sum_{i=0}^{n-1}\frac {t} nW_{\frac {t} ni}$ is also a Gaussian process. What about the limit as $n\rightarrow \infty$?
