Ergodic Process Selection

Let \(\{ ({{X}_{i}},{{Y}_{i}})\} _{{i = 1}}^{\infty }\) be a jointly ergodic stationary stochastic process. Define a selection function \({{\delta }_{n}}:{{X}^{{n - 1}}} \times {{Y}^{{n - 1}}} \to \{ 0,1\} ,n = 1,2, \ldots\) We wish to maximize $$\begin{array}{*{20}{c}} {\mathop{{\lim }}\limits_{{n \to \infty }} \frac{1}{n}\sum\limits_{{i = 1}}^{n} {({{\delta }_{i}}({{X}_{1}}, \ldots ,{{X}_{{i - 1}}},{{Y}_{1}},{{Y}_{2}}, \ldots ,{{Y}_{{i - 1}}}){{X}_{i}}} } \hfill \\ { + (1 - {{\delta }_{i}}({{X}_{1}}, \ldots ,{{X}_{{i - 1}}},{{Y}_{1}}, \ldots ,{{Y}_{{i - 1}}})){{Y}_{i}})} \hfill \\ \end{array}$$ over all selection functions. Thus δi chooses either Xi or Y i to add to the running average.