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Universal schemes for prediction, gambling and portfolio selection. (English) Zbl 0758.90006
This paper discusses universal schemes for portfolio selection. After the introduction to the problem a universal gambling scheme without the help of a universal prediction scheme is constructed. Then the relation between universal gambling schemes and universal modeling schemes are regarded. The universal modeling schemes regarded are mainly for the purpose of compression of individual sequences of finite length, and do not emphasize the asymptotic optimality of their schemes for stationary ergodic ensembles of random sequences. It follows a discussion of investment in the stock market. If 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 are 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. Further Ornstein’s universal prediction scheme is reviewed. It follows a discussion of a more general universal prediction scheme to learn, from past experience, the conditional distribution given in infinite past of next outcome of a stationary ergodic process with values in a Polish space.

MSC:
91G10 Portfolio theory
60G10 Stationary stochastic processes
47A35 Ergodic theory of linear operators
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