I want to show that the limit of: $$X_n(t)=\sum_{i=1}^{n}f_i(t)Y_i$$ namely $$X_\infty(t)=\sum_{i=1}^{\infty}f_i(t)Y_i$$ is a Gaussian process, where $f_i(t)$ is a deterministic function and $Y_i$ is a i.i.d. sequence of standard normal.
I'm reading the following proof that show that $c^TX$ is normally distributed for any $c\in\mathbb R^n$. $$\sum_{i=1}^{m}c_iX_{t_i}=\sum_{i=1}^{m}c_i\sum_{j=1}^{\infty}f_j(t)Y_j=lim_{n\to \infty}\sum_{j=1}^{n}Y_j\sum_{i=1}^{m}c_if_j(t)=lim_{n\to \infty}\sum_{j=1}^{n}a_jY_j$$
and it conclude saying that $lim_{n\to \infty}\sum_{j=1}^{n}a_jY_j$ is a linear combination of independent $N(0,1)$ and therefore it is normal and the limit too.
What do you think about the following proof and there is some other alternative? What i don't understand is for example how is made the vector X and why if in the first equality on the lhs we have $X_{t_i}$ then in the rhs it only depend on $j$ and no more on $i$. Moreover also the last eqaulity it is not clear to me. any help?
The argument is basically right but you have to change $t$ to $t_i$ in the inside sum $ \sum\limits_{j=1}^{\infty} f_j(t)Y_j$.