How to understand and derive the "density" related to likelihood ratio for some following linear model? I am struggling with one equation in the book "Asymptotic theory of statistical inference for time series". To be specific, I wonder how equation (2.2.9) on page 39 is derived (check here). To cut straight to the question, I will only state relevant notations and conditions.
Consider a model (equation (2.2.1), page 37):
$$
X(t) = \sum_{j=0}^\infty A_\theta(j) U(t-j)
$$ 
where $X(t)$ and $U(t)$ are $m$-random vectors, $\{U(t):t\in\mathbb{Z}\}$ is i.i.d., and $A_\theta(j)$ is a $m$-by-$m$ matrix with parameter $\theta$. Under some conditions (from page 37 to page 39), we may assume that the following representation holds (equation (2.2.6), page 38):
$$
U(t) = \sum_{j=0}^\infty B_\theta(j) X(t-j),\mbox{ where } B_\theta(0) = I_m
$$
This could be further represented as
$$
U(t) = \sum_{j=0}^{t-1} B_\theta(j) X(t-j) + \sum_{r=0}^\infty C_\theta(r,t) U(-r),
$$
with $C_\theta(r,t) = \sum_{r'=0}^{r} B_\theta(r'+t) A_\theta(r-r')$.
Now, denote by $p(\cdot)$ the probability density function of $U(t)$,  and let $Q_{n,\theta}$ and $Q_\mathbf{u}$ be the probability distributions of $\{U(s),s\leq 0, X(1), \dots, X(n)\}$ and $\{U(s), s \leq 0\}$ respectively.
My question: The authors then claim
\begin{equation}
d Q_{n,\theta} = \prod_{t=1}^{n} p\left(\sum_{j=0}^{t-1} B_\theta(j) X(t-j) + \sum_{r=0}^\infty C_\theta(r,t) U(-r)\right) d Q_\mathbf{u} \label{eq}
\end{equation}
I am really confused about this equation.
Question 1: What does the equation mean? It looks as if 
$$
\prod_{t=1}^{n} p\left(\sum_{j=0}^{t-1} B_\theta(j) X(t-j) + \sum_{r=0}^\infty C_\theta(r,t) U(-r)\right)
$$
is the Radon-Nikodym derivative of $Q_{n,\theta}$ w.r.t. $Q_\mathbf{u}$, but I cannot see why it makes sense, as $Q_\mathbf{u}$ and $Q_{n,\theta}$ are measures on different spaces.
Question 2: Why do we have random vectors $X(t)$ and $U(t)$ appearing inside $p(\cdot)$? I mean, $X(t)$ and $U(t)$ are defined on the (implicit) original probability space, but both $Q_{n,\theta}$ and $Q_\mathbf{u}$ are measures on some product space of $\mathbb{R}^m$, so I couldn't under what it means.
Question 3: Maybe I have misinterpreted the equation. Can anyone shed light on its derivation? It seems that we only need some change of variable here, but before deriving, I have to know what the equation means..
Any help will be greatly appreciated!
 A: That equation looks to me like an implementation of conditional probability rather than radon nikodym. I don't think this is the full answer but it might be a step in the right direction.
Notice that as knowing $X(1),\dots,X(n),U(s)$ tell us $U(t)$.
$$
dQ_{n,\theta} = P(U(s),X(1),\dots,X(t))\text{ tells us } P(U(t))
$$
\begin{align*}
dQ_{n,\theta} &=P(U(n)),\\
 &= P(U(n)|U(n-1))P(U(n-1)),\\
&=P(U(n)|U(n-1))P(U(n-1)|U(n-2))P(U(n-2)),\\
&=\prod_{t=2}^nP(U(t)|U(t-1)))P(U(1)),\\
&=\prod_{t=2}^nP(U(t)|U(t-1)))P(U(1)|U(s))P(U(s)),\\
&=\prod_{t=2}^nP(U(t)|U(t-1)))P(U(1)|U(s))dQ_u,\\
&=\prod_{t=1}^np(\sum_{j=1}^{t-1}B_\theta(j)X(t-j)+\sum_{r=0}^\infty C_\theta(r,t)U(-r))dQ_u,\\
\end{align*}
I do not think I've correctly interpreted the notation, but this is the closest I've come, it's based around thinking that maybe 
$$
p(\sum_{j=1}^{t-1}B_\theta(j)X(t-j)+\sum_{r=0}^\infty C_\theta(r,t)U(-r)) = P(U(t)|U(t-1))
$$
which may not be correct. I'm also not convinced that $dQ_{n,\theta} = P(U(n))$ but perhaps you might be able to adapt this argument in some way. I hope this helps.
