Question About Kailath's Paper - An Innovations Approach to Least Squares Estimation Part I: Linear Filtering in Additive White Noise I'm reading the paper at the link below and I was following it for about 2 pages until I hit a road block on the bottom of page 648 where the author says: 


  putting together 9-11, we obtain


and  gives  equation 12. I understand equations 9, 10 and 11  but don't see how 12 comes about from them. 

Also, if anyone knows a book or a paper that provides an exact mathematical definition of white noise, it's appreciated. Definitely, I'm seeing how important it is to understand white noise, particularly in the context of this paper. Gaussanity ( which is what I'm used to seeing  in the many derivations of the Kalman filter ) is clearly quite different ( Dilip, I always believed you but now I'm seeing the importance )  and doesn't cut it here. This paper is actually the first one of a four part-er but, if I can follow this one by the end of 2018, I'd be quite satisfied.

An Innovations Approach to Least Squares Estimation Part I: Linear Filtering in Additive White Noise.
They used to have different convention for writing this stuff back then.
But actually what you saw is really simple.  

It's all based on the Orthogonal Principle of MMSE.

They say in (9) that the Additive Noise is uncorrelated (By defining its Auto Correlation by Delat Function).
In (10) they say the optimal estimation of $ x \left( t \right) $ is given by $ \hat{x} \left( t \mid t \right) $ is a linear combination of $ y \left( t  \right) $ and $ z \left( t \right) $.
In (11) they exactly use the Orthogonal Principle which the estimator must obey.  

In (12) they just derive the Correlation Matrix between the processes.
Since the linear estimator represent the correlation it is not surprising to see that it is the linear combination defined by the optimal filter $ g \left( \cdot \right) $.

The trick here is the fact (Using article notations) $ \overline{ \left( x \left( t \right) - \hat{x} \left( t \mid t \right) \right) v' \left( s \right) } = 0 $  which means (By Linearity of the Expectation) $ \overline{ x \left( t \right) v' \left( s \right) } = \overline{ \hat{x} \left( t \mid t \right) v' \left( s \right) } $.
Expand the right hand term by (10) by plugging $ \hat{x} \left( t \mid t \right) $ and you get (12).

By the way, Kaliath has much better written, in my opinion, paper on the subject called A View of Three Decades of Linear Filtering Theory.