TITLE: On convergence of stochastic processes
QUESTION [1 upvotes]: This is a question about convergence of stochastic processes.
Let $\{X^{(n)}\}_{n=1}^\infty=\{X^{(n)}_t, t \in [0,1]\}_{n=1}^\infty$ be a sequence of continuous-time stochastic processes satisfying the following properties:

Each $X^{(n)}$ is a real-valued diffusion process.
The laws $\{P^{(n)}\}_{n=1}^\infty$ of $\{X^{(n)}\}_{n=1}^\infty$ are tight in $C([0,1],\mathbb{R})$. Here $C([0,1],\mathbb{R})$ denotes the space of real-valued continuous functions on $[0,1]$ equipped with the sup norm.
There exists a subsequence of $\{P^{(n)}\}_{n=1}^\infty$ which converges weakly in $C([0,1],\mathbb{R})$ to a one-dimensional Brownian motion.

Then, can we show that $\{P^{(n)}\}_{n=1}^\infty$ converges weakly in $C([0,1],\mathbb{R})$ to a one-dimensional Brownian motion?

REPLY [1 votes]: I think the answer is no.
Easy counterexample: let $B$ be a brownian motion and $X^{(n)}_t=0$ for $n$ even, and $X^{(n)}_t=B_t$ for $n$ odd.
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