= Solution
Finite independent Gaussian sums are Gaussian; directly, multiplication of their <characteristic functions> gives
$$
\varphi_{Y_n}(t)=\prod_{j=1}^ne^{-\alpha_j^2t^2/2}=\exp\left(-\frac{t^2}{2}\sum_{j=1}^n\alpha_j^2\right)\longrightarrow e^{-\sigma^2t^2/2}.
$$
By <Markov inequality>, $\mathbb P(|Y_n-Y|>\epsilon)\leq\epsilon^{-2}\|Y_n-Y\|_2^2\to0$. Part (ii) makes the same sequence converge in distribution to $Y$, and the characteristic-function estimate there makes $\varphi_{Y_n}\to\varphi_Y$. Uniqueness of distributions determined by characteristic functions therefore yields
$$
\boxed{Y\sim N(0,\sigma^2).}
$$
If $\sigma^2=0$, all coefficients vanish and $Y=0$ almost surely; $N(0,0)$ here denotes that degenerate distribution.
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