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Random arithmetic-geometric means and random pi: Observations and conjectures. (English) Zbl 0756.60056
Two random versions of the arithmetic-geometric mean of Gauss, Lagrange and Legendre are defined. Almost sure convergence and nondegeneracy are proved. These random arithmetic-geometric means in turn define two random versions of \(\pi\). Based on numerical simulations, inequalities and equalities are conjectured. A special case is proved. The authors consider the work as a first step in a more general study of nonlinear iterations with random parameters, heading towards a possible marriage between nonlinear dynamical systems and stochastic processes.
MSC:
60J05 Discrete-time Markov processes on general state spaces
65D20 Computation of special functions and constants, construction of tables
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