This question seems to be related to Theorem IX.7.28 in J. Jacod and A. Shiryaev's *Limit theorems for stochastic processes* (2013), and it is very important to prove asymptotic properties of my statistical estimator.
The situation can be much like simplified as follows.

**Here is a question.**

Let our discretized process be \begin{align} X^n_t = \sum_{i=1}^{[nt]} \left(W_{\frac{i}{n}} - W_{\frac{i-2}{n}}\right), \end{align} where $W$ is a standard Wiener process. My intuition says that it would converge stably in law to $2W$ (in the Skorokhod space $\mathbb{D}([0,1])$), i.e., \begin{align} X^n \xrightarrow{\mathcal{L}-s} 2W. \end{align} But I'm not sure it is ok to apply Theorem IX.7.28 to show this, because the predictable part (denote it as $B^n_t$) converges to Wiener process itself which has "infinite variation": \begin{align} B^n_t &:= \sum_{i=1}^{[nt]} \mathbb{E}_{\frac{i-1}{n}} \left[ W_{\frac{i}{n}} - W_{\frac{i-2}{n}} \right] \\ &= \sum_{i=1}^{[nt]} \left(W_{\frac{i-1}{n}} - W_{\frac{i-2}{n}}\right) \\ &= W_{\frac{[(n-1)t]}{n}} \xrightarrow{\mathbb{P}} W_t, \end{align} while Jacod and Shiryaev's theorem says that $B^n_t$ should converge to a predictable "finite variation" process $B_t$. So how can I show this?

**Here is a theorem.**

Without considering the truncation function or jumps of $X$, the theorem says:

IX.7.28 Theorem.For every càdlàg process $X$, we use the following notation: \begin{align*} X_t^n = X_{[nt]/n},\qquad\qquad \Delta_i^n X = X_{i/n} - X_{(i-1)/n} = \Delta X^n_{i/n}, \end{align*} We also consider the discretized process of the form \begin{align*} X_t^n = \sum_{i=1}^{[nt]} \chi_i^n, \end{align*} where each $\chi_i^n$ is $\mathcal{F}_{i/n}$-measurable. Assume that each $\chi_i^n$ is square-integrable, and $X$ is a continuous and $\mathbb{E}[|X_t|^2] < \infty$ for all $t$ with the canonical decomposition $X_t = B_t + M_t$ where $B_t$ is predictable finite variation process and $M_t$ is square-integrable local martingale.Suppose also that for all $t>0$ and all uniformly integrable martingale $N$ which are orthogonal to $X$ we have \begin{align*} \sup_t \left| \sum_{i=1}^{[nt]} \mathbb{E} _{\frac{i-1}{n}}[\chi_i^n] - B_t \right| &\xrightarrow{\mathbb{P}} 0, \\ \sum_{i=1}^{[nt]} \mathbb{V}_{\frac{i-1}{n}}[\chi_i^n] &\xrightarrow{\mathbb{P}} \langle M, M \rangle_t + \langle w \cdot W', w \cdot W' \rangle_t \\ \sum_{i=1}^{[nt]} \mathbb{E}_{\frac{i-1}{n}}[\chi_i^n \Delta_i^n M] &\xrightarrow{\mathbb{P}} \langle M, M \rangle_t, \\ \sum_{i=1}^{[nt]} \mathbb{E}_{\frac{i-1}{n}}[\chi_i^n \Delta_i^n N] &\xrightarrow{\mathbb{P}} 0. \end{align*} Then there is a very good canonical Wiener extension of $(\Omega, \mathcal{F}, \{\mathcal{F}_t\}, \mathbb{P})$ with Wiener process $W'$ and a continuous $X$-biased $\mathcal{F}$-progressive conditional martingale PII $X'$ on this extension such that \begin{align*} X^n \xrightarrow{\mathcal{L}-s} X' = X + w \cdot W', \end{align*} where $w \cdot W' = \int w\,dW'$ and $w$ is a predictable process.

If I have any misunderstanding please let me know. Any help will be appreciated.

Thanks,

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