Independent increments of a Gaussian martingale

A centered continuous Gaussian process that is also a martingale has independent increments. Every future increment has zero covariance with every finite vector of past values by the martingale property; uncorrelated jointly normal variables are independent.

Every centered continuous Gaussian martingale starting at zero is a deterministic time change of Brownian motion. Its clock is the continuous increasing function $f(t)=\mathbb{E}[M_t^2]$, and $[M{]}_t=f(t)$.

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.