Let $S_t$ be the Geometric Brownian Motion, we know that $$dS_t=rS_tdt+\sigma S_tdW_t, t\in [0,T], S_0>0, r>0,\sigma>0$$ and the distribution of $S_t$ is known explicitly. Please see the link above for the density function of $S_t$. Let $h(.)$ be a bounded continuous function (as nice as you want), and consider the discretely monitored $S_{t_1},S_{t_2},\ldots, S_{t_n}$ with log-normal distribution of $S_t$. Assume that we have $$S^{k}_{t_i}\Longrightarrow S_{t_i},\quad\text{weakly as}\quad k\to\infty,$$ where $S_{t_i}$ is log-normally distributed.

**My question here** : Do we have the following weak convergence
$$
\displaystyle\sum_{i=1}^{n}h(S^{k}_{t_i})
\Longrightarrow \displaystyle\sum_{i=1}^{n}h(S_{t_i}),\quad\text{as}\quad k\to\infty?
$$
In other words, do we have
$$(S_{t_1}^k,\ldots,S_{t_n}^k)\Longrightarrow (S_{t_1},\ldots,S_{t_n}) \quad\text{as}\quad k\to\infty?$$
We note the above is not true in general but I am not sure for the log-normal random variable case. It would be nice if some one could give me some ideas or hints. Thank you so much for your time. I truly appreciate it.