I am reading a paper and having trouble with understanding the following. The setup is two stochastic processes, $(X_t)_{t\ge 0}$ and $(W_t)_{t\ge 0}$, where $(W_t)_{t\ge 0}$ is just a Brownian motion. It is assumed we can redefine both of these on a new probability space so that the joint distributions coincide.
Let $(Y_n)_{n\in \mathbb{N}_0}$ be a discrete process and suppose that there exists non-negative random variables $T_k$ such that
$\{W(\sum_{j\le M}T_j),M\ge 1\}$ and $\{\sum_{j\le M}Y_j,M\ge 1\}$
have the same distribution (Skorokhod representation). Then $(Y_n)$ is redefined on a new probability space by
$Y_M=W(\sum_{j\le M}T_j)-W(\sum_{j<M}T_j)$.
For an $M\ge 1$ and define the process $S(t)=\sum_{j=1}^{M[t]}Y_j$.
The paper then says that following, which I do not understand:
We can redefine $(X_t)_{t\ge 0}$, $(S_t)_{t\ge 0}$, and $(W_t)_{t\ge 0}$ on another probability space so that the joint distribution of $(S_t)_{t\ge 0}$ and $(W_t)_{t\ge 0}$ and the joint distribution of $(X_t)_{t\ge 0}$ and the old version of $(S_t)_{t\ge 0}$ remain unchanged. Simply choose the newer probability space in such a way that $(X_t)_{t\ge 0}$ and $(W_t)_{t\ge 0}$ are conditionally independent given $(S_t)_{t\ge 0}$.
Could anyone care to elaborate on this?
The paper is "Almost sure invariance principles for partial sums of weakly dependent random variables" by Walter Philipp and William Stout.