= Gaussian series with square-summable coefficients
{c}
{title2=$\sum_j\alpha_jX_j\sim N(0,\sum_j\alpha_j^2)$}
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.
Back to article page
