F-Brownian bridge (${\mathbb{G}}_F(t)=\mathbb{G}(F(t))$)

For a cumulative distribution function $F$ and a standard Brownian bridge $\mathbb{G}$, the F-Brownian bridge is the centred Gaussian process ${\mathbb{G}}_F=\mathbb{G}\circ F$. Its covariance is $F(\min (s,t))-F(s)F(t)$, by substituting $F(s),F(t)$ into the bridge covariance. It can jump when $F$ jumps. Its law is tight in the supremum norm, since composition with $F$ is a continuous map from the separable space of continuous bridge paths.