In my previous post I asked the following problem (to which I have received convincing replies, which I summarize below with the statement of the problem):
If $X$ and $Y$ are stochastic processes such that $X_t\stackrel{d}{=}Y_t$ for all $t\geq 0$, this is not enough to guarantee that $\mathcal{L}(X)=\mathcal{L}(Y)$ (where $\mathcal{L}(X)$ indicates the law of the stochastic process $X$).
My new problem is the following. Suppose that $\{\xi_{j,n}| j=1,\dots,n,n\in\mathbb{N}^{\star}\}$ is a triangular array of random variable such that $\xi_{j,n}\stackrel{d}{=}u_{j}$ for all $n$ and for all $j=1,\dots,n$, where $u_j$ are iid random variables. Define $$ Y_t^{(n)} \doteq \sum_{j=1}^{\lfloor n\,t\rfloor}\xi_{j,n},\quad X_t^{(n)} \doteq \sum_{j=1}^{\lfloor n\,t\rfloor}u_{j} $$ Can I conclude that $\mathcal{L}(Y^{(n)})=\mathcal{L}(X^{(n)})$ ?
An attempt. My guess is that the Lemma VI.3.19 of Jacod and Shiryaev's book can be now applied. This is the Lemma:
Let $X$ and $Y$ be two cadlag stochastic processes. If $$ \mathcal{L}(Y_{t_1},\dots,Y_{t_k}) = \mathcal{L}(X_{t_1},\dots,X_{t_k}) $$ for all $t_j\in D$, $D$ dense subset of $\mathbb{R}_{+}$, and for all $k\in\mathbb{N}^{\star}$, then $\mathcal{L}(X)=\mathcal{L}(Y)$.
So the idea is the following: no matter how I take $t_{1},\dots,t_{k}$ in $\mathbb{R}_{+}$ we will have $$ Y^{(n)}_{t_{j}}=\sum_{\ell=1}^{q(j)}\xi_{\ell,n},\quad X^{(n)}_{t_{j}}=\sum_{\ell=1}^{q(j)}u_{\ell} $$ for some mapping $j\in\{1,\dots,k\}\rightarrow q(j)\in\mathbb{N^{\star}}$, so that (I need to resort to image attaching since I cannot let the tex compiler work properly here)
Addendum. To better motivate the problem, this is where it typically comes from. Suppose that you have a function $f$ regular enough and let $W$ be a Brownian motion. You want to study the convergence in law of the sequence of processes $$ Y_t^{(n)} = \Delta_n\sum_{j=1}^{\lfloor t/\Delta_n\rfloor}f\left(\frac{W_{j\,\Delta_n}-W_{(j-1)\,\Delta_n}}{\sqrt{\Delta_n}}\right) $$ where $\Delta_n\rightarrow 0$ as $n\rightarrow+\infty$. To do that one exploits the fact that $$ \xi_{j,n}\doteq f\left(\frac{W_{j\,\Delta_n}-W_{(j-1)\,\Delta_n}}{\sqrt{\Delta_n}}\right) \stackrel{d}{=} u_j $$ where $u_j$ are iid random variables (as the $\xi_{j,n}$ are), but, most importantly, they are not a triangular array (this is why, inside the function, the normalization $\sqrt{\Delta_n}$ is chosen). In particular the $u_j$ have a Gaussian distribution with mean $\rho(f)$ and variance $\rho(f^2)-\rho(f)^2$ with $\rho(f)=\int_{\mathbb{R}}f(x)\mu(dx)$ where $\mu$ is the standard Gaussian measure.