Ergodic Process Selection

The purpose of this note is to give a partial solution to the following problem posed by Thomas M. Cover [1]. Let (X, Y) = (X_i, Y_i, i ∈ Z) be a jointly ergodic stationary stochastic process. A random process δ = (δ_i, i ∈ Z) is called a selection strategy if δ_i, ∈ {0,1} with probability one for each i, and a selection strategy δ is called sequential if for each i ≥ 1, δ_i is measurable with respect to $$\sigma ({{X}_{{i - 1}}},{{Y}_{{i - 1}}},{{X}_{{i - 2}}},{{Y}_{{i - 2}}}, \ldots ,{{X}_{1}},{{X}_{1}}),$$ which represents the finite past.