 Solution

Finite independent Gaussian sums are Gaussian; directly, multiplication of their characteristic functions givesBy Markov inequality, $\mathbb{P}(|Y_n-Y|>ϵ)\le {ϵ}^{-2}\Vert Y_n-Y{\Vert }_2^2\to 0$. Part (ii) makes the same sequence converge in distribution to $Y$, and the characteristic-function estimate there makes ${\phi }_{Y_n}\to {\phi }_Y$. Uniqueness of distributions determined by characteristic functions therefore yieldsIf ${\sigma }^2=0$, all coefficients vanish and $Y=0$ almost surely; $N(0,0)$ here denotes that degenerate distribution.