# The Radon-Nikodym derivative for a sequence of dependent variables

Suppose that a probability space $$(\Omega, \Sigma, \mathbb{P})$$ is given. Let $$W=\{W_n\}_{n\in \mathbb{N}_0}$$ be a sequence of $$\mathbb{P}$$-i.i.d real-valued random variables on $$\Omega$$. Furthermore, assume that $$\mathbb{Q}$$ be a new probablity measure on $$\Sigma$$, and $$W$$ is $$\mathbb{Q}$$-i.i.d and that $$\mathbb{Q}_{W_1} \sim \mathbb{P}_{W_1}$$. We denote by $$\mathcal{F}^{W} = \{\mathcal{F}_n^W\}_{n\in \mathbb{N}}$$ the natural filtration of $$W$$. Then, one can say that for every $$n\in \mathbb{N}_0$$ and for all $$D\in \mathcal{F}_n^W = \sigma(W_1, W_2, .., W_n)$$ $$\begin{equation} \mathbb{Q}(D) = \mathbb{E}^{\mathbb{P}}\Big[I_D \prod_{j=1}^{n}\frac{d\mathbb{Q}_{W_1}}{d\mathbb{P}_{W_1}}(W_j)\Big] \end{equation}$$ or equivalently, $$\begin{equation} \frac{d\mathbb{Q}}{d\mathbb{P}}\Big|_{\mathcal{F}_n^W} = \prod_{j=1}^{n}\frac{d\mathbb{Q}_{W_1}}{d\mathbb{P}_{W_1}}(W_j) \end{equation}$$ Now, my question is what if $$W_j's$$ is not independent. In this case, how does the Radon-Nikodym derivative $$\frac{d\mathbb{Q}}{d\mathbb{P}}\Big|_{\mathcal{F}_n^W}$$ look like?