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?
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