has to be carried out so that this partitioning is preserved.
Since this partitioning is arbitrary (as far as the mathematics is
concerned), it is obvious that a space which is not partitioned will
have many (equivalent) decompositions. On the other hand, if the
partitioning is into more than two parts, then the existence of a
decomposition is not guaranteed.
A slight penalty has to be paid for the use of this partitioning,
namely: instead of eventually obtaining a random cartesian product of
one-dimensional spaces, we obtain an extended channel (with random
input) of single-dimensional channels. It is obvious that if we were
to drop the partitioning temporarily, each such single-dimensional
channel would be further decomposed into two random components. This
decomposition is not unique. But one of these equivalent decompositions
is particularly convenient; namely, that decomposition where we take
the component out of the original X′ and that which is random to it,
say V. This V (as well as the cartesian product of all such V’s,
which of necessity are random) is called the linearly additive noise.
The name “linearly additive” is justified because it is just the
statistical concept isomorphic to the linear addition of vectors in
orthogonal Euclidean geometry. (The proof of this last statement is not
completed as yet.)