Proposition: Repetition preserves static task information

For $T\ge 1$, let

Then

More generally, if $Y⟶W⟶O$ is a Markov chain, then $Q^{\ast }(O)\le Q^{\ast }(W)$.

Proof. The repeated sequence is a function of $X$, and its first coordinate recovers $X$. Prediction rules can therefore be translated in either direction. For the Markov claim, simulate the conditional law of $O$ given $W$ and then apply any rule defined on $O$.