Given a $T>0$, let $\mathcal{C}[0,T]$ be the space of continuous functions on $[0,T]$. Let $Y_n(t)$ be stochastic processes in $\mathcal{C}[0,T]$. We define the weak convergence in the sense of uniform metric and denote it as "$\Rightarrow$", see Billingsley (1968) Convergence of Probability Measures, Chapter 2.
Suppose we have the joint convergence $$ (X_n,Y_n)\Rightarrow (X,c) \quad \text{as} \quad n\to\infty, $$ where $X_n$ and $X$ are random variables, $Y_n\in\mathcal{C}[0,T]$ and $c$ is a "constant" process.
Can we prove that $$ (X_n,Z_n)\Rightarrow(X,Z) \quad \text{as} \quad n\to\infty, $$ where $Z_n(t)=\int_0^t Y_n(s){\rm d}B(s)$ are stochastic integrals and $Z(t)=cB(t)$, and $B$ is a standard Brownian motion being independent of $X_n$.
Notice that, the weak convergence theory of stochastic integrals do show that $Z_n\Rightarrow Z$, but I don't find an answer for the above joint weak convergence.
Could you provide a detailed proof ? Or provide a counter-example ?
Thank you !