Gaussian process construction from square-summable features ($X_t={\sum }_i{k}_i(t){\xi }_i,K(s,t)={\sum }_i{k}_i(s)k_i(t)$)

For independent standard normal variables ${\xi }_i$ and ${\sum }_i{k}_i(t{)}^2<\infty$ at each fixed time, the series defines $X_t$ in $L^2$. Every finite linear combination is an $L^2$ limit of Gaussian variables and is Gaussian. Thus this constructs a Gaussian process with the Gram covariance kernel $K$. It supplies no sample-path continuity without additional assumptions.