The initial efficiency of investment for the general market

We investigate how one invests in the stock market X when there is a rate constraint on the side information V. The doubling function Δ(R) is the maximum increase in the doubling rate when V is described to the investor at rate R. The initial efficiency Δ´(0) is the largest possible increase in the doubling function per bit of description. We introduce the linearized maximal correlation and use it to provide a lower bound for the initial efficiency. We also show that the initial efficiency is bounded above by the square of the Hirschfeld-Gebelein-Renyi maximal correlation between the side information V and the market X. We can use these bounds to find the initial efficiency for V=X and for the horse race market