The relation of description rate and investment growth rate

We have shown that if one invests in the outcome of a random variable X, where investment consists of gambling at any odds, then every bit of description of X increases the doubling rate by one bit. However, if the provider of the information has access only to V, a random variable jointly distributed with X, then this maximal efficiency is not generally possible. We find the increase Δ(R) in doubling rate for a description of V at rate R for the jointly Gaussian and jointly binary cases. We investigate the extension to multivariate Gaussian random variables. We prove a general result for the derivative of Δ(R) at R=0. We then consider the problem in which there are k separate encoders and each observes a random variable V_i correlated with X. We find how efficiently these encoders, without cooperation, help the investor who is interested in X