Let B a Brownian motion. I want to show that What I want to show is that $$\int_0^tf(s)\,dB_s$$ is a gaussian process, where f(t) is a deterministic and continuous function. From the definition of stochastic process we know: $$ \sum_{i=1}^{K_n-1}f(s)(B_{t_{i}^n}-B_{t_{i-1}^n}) \to \int_0^tf(s)\,dB_s=X_t$$ I can observe that $\sum_{i=1}^{K_n-1}f(s)(B_{t_{i}^n}-B_{t_{i-1}^n})$ is a sum of independent normal random variable and therefore is still Normal. Moreover also the limit of normal r.v. is still normal. I can then conclude that $\int_0^tf(s)\,dB_s$ is a Normal random variable.
Now is coming the part where I have some doubt: Can i conclude that vec$(X_{t_1},.. X_{t_n})$ is multivariate normal saying that each $X_{t_1},.. X_{t_n}$ is an independent normal random variable? I'm not sure about the independence. It seems to me that they are independent since every $X_{t_1}$ is the result of a sum of independent random variable.
Any help please?