Beyond the Central Limit Theorem: Universal and Non-universal Simulations of Random Variables by General Mappings
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The Central Limit Theorem states that a standard Gaussian random variable can be simulated within any level of approximation error (measured by the Kolmogorov-Smirnov distance) from an i.i.d. real-valued random vector X^n∼ P_X^n by a normalized sum mapping (as n→∞). Moreover given the mean and variance of X, this linear function is independent of the distribution P_X. Such simulation problems (in which the simulation mappings are independent of P_X, or equivalently P_X is unknown a prior) are referred to as being universal. In this paper, we consider both universal and non-universal simulations of random variables with arbitrary target distributions Q_Y by general mappings, not limited to linear ones. We derive the fastest convergence rate of the approximation errors for such problems. Interestingly, we show that for discontinuous or absolutely continuous P_X, the approximation error for the universal simulation is almost as small as that for the non-universal one; and moreover, for both universal and non-universal simulations, the approximation errors by general mappings are strictly smaller than those by linear mappings. Specifically, for both universal and non-universal simulations, if P_X is discontinuous, then the approximation error decays at least exponentially fast as n→∞; if P_X is absolutely continuous, then only one-dimensional X is sufficient to simulate Y exactly or arbitrarily well. For continuous but not absolutely continuous P_X, using a non-universal simulator, one-dimensional X is still sufficient to simulate Y exactly, however using a universal simulator, we only show that the approximation error decays sup-exponentially fast. Furthermore, we also generalize these results to simulation from Markov processes, and simulation of random elements (or general random variables).
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