Risk-sensitive probability for Markov chains

The probability distribution of a Markov chain is viewed as the information state of an additive optimization problem. This optimization problem is then generalized to a product form whose information state gives rise to a generalized notion of probability distribution for Markov chains. The evolution and the asymptotic behavior of this generalized or “risk-sensitive” probability distribution is studied in this paper and a conjecture is proposed regarding the asymptotic periodicity of risk-sensitive probability and proved in the two-dimensional case. The relation between a set of simultaneous non-linear and the set of periodic attractors is analyzed.