Innovations sequence

Definition 
Suppose $\{z_k\}$ is a sequence of jointly Gaussian random elements.   The innovations process $\{\tilde{z}_k\}$ is such that $\tilde{z}_k$ consists of that part of $z_k$ containing new information not carried in $z_{k-1}, z_{k-2}, \dotsc$.
$$\tilde{z}_k = z_k - E[z_k | z_0, \dotsc, z_{k-1} ]  = z_k - E[z_k | Z^{k-1}]$$ with $\tilde{z}_0 = z_0 - E[z_0]$.

Properties

1. $\tilde{z}_k$ independent of $z_0, \dotsc, z_{k-1}$ by definition
2. (1) implies $E[ \tilde{z}_k' \tilde{z}_l] = 0, l \neq k$
3. $E[z_k | Z^{k-1}]$ is a linear combination of $z_0, \dotsc, z_{k-1}$
4. The sequence $\{\tilde{z}_k\}$ can be obtained from $\{z_k\}$ by a causal linear operation.
5. The sequence $\{z_k\}$ can be reconstructed from $\{\tilde{z}_k\}$ by a causal linear operation. 
6. (4) and (5) implies $E[z_k | Z^{k-1}] = E[z_k | \tilde{Z}^{k-1}]$ or more generally  $E[w | Z^{k-1}] = E[w | \tilde{Z}^{k-1}]$ for jointly Gaussian $w, \{z_k\}$
7. For zero mean Gaussian $\tilde{x}_k$, $\tilde{z}_k$, we have  $$E[x_k|Z^{k-1}] = E[x_k|\tilde{Z}^{k-1}] = E[x_k| \tilde{z}_0] + \dotsb + E[x_k| \tilde{z}_{k-1}]$$