Random number theory and real analytic probability

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Before our project, it had been proved that dependence of almost all irrational rotation vanish when it applied for the central limit theorem of Rademacher functions. This result explains why the quasi Monte Carlo methods are not effective in high dimensional space comparing with the ordinary Monte Carlo method.Concerning this result, we classified irrational rotations and studied in detail on irrational rotations which generate the sequence converges to dependent stationary sequence. We have given conditions to have independent limit sequence and succeeded to have the numerically effective mean to verify the independence of limit.We prove the central limit theorem for multiple n-ary transforms. This results is closely related to the theory of accelation of generation of random numbers by means of multiple transforms.