I've one question regarding the convergence of the sum of two stochastic processes. Let $(X^n_{t})_t \rightarrow (X_t)_t$ and $(Y^n_{t})_t \rightarrow (Y_t)_t$ for $n \to \infty$ where $\rightarrow$ denotes convergence in distribution in the uniform metric in $C[0,\infty)$. Assume that $X^n_t$ and $Y^n_t$ are independent as well as $X_t$ and $Y_t$ are independent. Does $(X^n_t+Y^n_t)_t \rightarrow (X_t+Y_t)_t$ hold? I know that the statement is true in fdd convergence but does it also hold if one regards the processes with the uniform metric?
Many thanks in advance!