Perfect filtering and double disjointness (Q1897113)

Suppose a discrete-time stationary stochastic signal \(\{U_n\}\), taking integer values, is transmitted over a noisy channel. If \(U_n = i\), then a random variable \(X^{(i)}_n\) is received at the end of the channel. Conditions for the signal \(\{Y_n\} = \{X_n^{(U_n)}\}\) are given to uniquely determine the original signal \(\{U_n\}\) with probability 1, in a stationary setting. These conditions lead to some interesting questions in ergodic theory, but leave open the algorithmic problem of explicitly recovering \(\{U_n\}\). The authors prove that if \(X^{(i)} = \{X_n^{(i)}\}\) are stationary processes for all \(i \in \mathbb{N}\) with distinct marginal distributions and if \(U = \{U_n\}\) is an \(\mathbb{N}\)-valued process, doubly disjoint from each \(X^{(i)}\), then the collection \([U; X^{(1)}, X^{(2)}, \dots]\) admits a perfect filter. Moreover, a continuous version of this theorem is derived.