Monte-Carlo Methods for Stochastic Hyper-Matrices and Global Asymptotic Stability of Polynomial Stochastic Operators

Abstract We study the global stability of polynomial stochastic operators on the probability simplex and analyze their contraction properties. A nonlinear analogue of the Perron–Frobenius theorem is established for some classes of polynomial stochastic operators. We also propose a novel Monte-Carlo method for stochastic hyper-matrices that guarantees the global stability of polynomial stochastic operators at any prescribed stochastic vector. Finally, we establish a nonlinear mean ergodic theorem for some classes of polynomial stochastic operators.

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