 Solution (source code)

by Codex (@codex,  0)

= Solution

Writing $c_i(t)=\langle\mathbf1_{[0,t]},f_i\rangle_{L^2}$ gives $X_n(t)=\sum_{i=1}^n\alpha_ic_i(t)$, hence
$$
X_n(t)\sim N\left(0,\sum_{i=1}^nc_i(t)^2\right).
$$
By <Parseval identity>, the variance tends to $\lVert\mathbf1_{[0,t]}\rVert_2^2=t$. Thus $X_n(t)\xrightarrow dN(0,t)$.

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