I am interested in weak convergence of stochastic processes with sample paths in $D_{\mathbb{H}}[0,1]$.
Let $\mathbb{H}$ denote a separable Hilbert space and $D_{\mathbb{H}}[0,1]$ the space of right continuous functions $x:[0,1] \to \mathbb{H}$ with left limits. Let $(X_{n})_{ n \geq 1} , X$ be random elements in $D_{\mathbb{H}}[0,1]$. When does $X_{n} \overset{D}{\to} X$ hold?
If $\mathbb{H}$ are the real numbers, there is a result in Billingsley 1968 (Theorem 15.6) requiring:
$X_{n} \overset{f.d.d.}{\to} X$ and
$E|X_{n}(t)-X_{n}(t_{1})|^{\gamma}|X_{n}(t_{2})-X_{n}(t)|^{\gamma}\leq (F(t_{2})-F(t_{1}))^{2\alpha}$ for $t_{1} \leq t \leq t_{2}$ $\gamma \geq 0$ $\alpha > \frac{1}{2}$
I have seen results in $D_{\mathbb{H}}[0,1]$ in Ethier and Kurtz but my question is: Is there a generalisation of Billingsleys result interchanging the absolute values by the norm of $\mathbb{H}$?