Deterministic quadratic variation characterizes a Gaussian continuous local martingale

If a continuous local martingale starts at zero and has deterministic continuous quadratic variation $[M{]}_t=f(t)$, the Dambis-Dubins-Schwarz theorem gives $M_t=B_{f(t)}$. It is therefore a centered Gaussian process. The converse follows from independent increments of a Gaussian martingale.