 Gaussian series with square-summable coefficients

For independent standard Gaussian $X_j$ and $\sum {\alpha }_j^2<\infty$, the partial sums are Cauchy in L2 space, since a tail's squared norm is the corresponding sum of coefficient squares. Completeness gives an $L^2$ limit. The finite-sum characteristic functions converge to $\exp [-t^2\sum {\alpha }_j^2/2]$, identifying the limit as Gaussian, including the zero-variance case. The identification needs convergence of distributions, not just agreement of the first two moments.