The relation of description rate and investment growth rate

摘要

We have shown that if one invests in the outcome of a randomvariable X, where investment consists of gambling at any odds, thenevery bit of description of X increases the doubling rate by one bit.However, if the provider of the information has access only to V, arandom variable jointly distributed with X, then this maximal efficiencyis not generally possible. We find the increase Δ(R) in doublingrate for a description of V at rate R for the jointly Gaussian andjointly binary cases. We investigate the extension to multivariateGaussian random variables. We prove a general result for the derivativeof Δ(R) at R=0. We then consider the problem in which there are kseparate encoders and each observes a random variable V_icorrelated with X. We find how efficiently these encoders, withoutcooperation, help the investor who is interested in X