Universal Schemes for Prediction, Gambling and Portfolio Selection

We discuss universal schemes for portfolio selection. When such a scheme is used for investment in a stationary ergodic market with unknown distribution, the compounded capital will grow with the same limiting rate as could be achieved if the infinite past and hence of the distribution of the market were known to begin with. By specializing the market to a Kelly horse race, we obtain a universal scheme for gambling on a stationary ergodic process with values in a finite set. We point out the connection between universal gambling schemes and universal modeling schemes that are used in noiseless data compression. We also discuss a universal prediction scheme to learn, from past experience, the conditional distribution given the infinite past of the next outcome of a stationary ergodic process with values in a Polish space. This generalizes Ornstein's scheme for finite-valued processes. Although universal prediction schemes can be used to obtain universal gambling and portfolio schemes, they are not necessary.