I'm interested in the following process on the space of $d \times d$ real valued matrices, $M_d(\mathbb{R})$.
Fix $n \in \mathbb{N}$ and consider the process $$X_{k,n} = \left( I + \frac{\epsilon_{k,n}}{\sqrt{n}} U_{k,n} \right) X_{k-1,n} \quad k=1,\dots n$$ with $X_{0,n} = I$ (identity). Here $\epsilon_{k,n}$ are i.i.d with finite fourth moment and $U_{k,n} \in M_d(\mathbb{R})$ with uniformly bounded (in $k$ and $n$) matrix norm. Now define a continuous time matrix valued process via $$ X_n(t) = X_{\lfloor nt \rfloor, n} \text{ for } t \in[0,1]. $$
I think of this as a vector valued process in $\mathbb{R}^{d^2}$. I would like to show that the sequence $X_n(t)$ is tight. A standard way might be to show that that there is a $C < \infty$ such that for every $\ell, k, n$, $$ \mathbb{E} || X_{k,n} - X_{\ell,n} ||^4_4 \leq C \left( \frac{k-\ell}{n} \right)^2, $$ where $||A||_4$ is the entrywise matrix matrix norm. Ie., $||A||_4^4 = \sum_{i,j} (A_{ij})^4$. This would imply tightness of each entry of the matrix process.
This way of doing such a calculation is quite messy. It is easy to calculate things like $$ \mathbb{E} || X_{k+1,n} - X_{k,n} ||^4_4, $$ via the recursion but since there is no triangle inequality this does not seem to help me. Any tips would be much appreciated. I'm also sure that this type of process has been studied before and so a reference would be a big help. Thanks.