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Beyond the Central Limit Theorem : Universal and Non-universal Simulations of Random Variables by General Mappings

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 ∼ P X by the normalized sum 1 Var(X)n ∑n i=1 (Xi − E [X]) (as n→∞). Moreover given the mean E [X] and variance Var(X), this linear function does not depend on the distribution PX . Such simulation problems (in which the simulation mapping is independent or almost independent of PX , or equivalently PX 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 an arbitrary target distribution QY 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 PX , 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 the seed distribution PX is discontinuous, then the approximation error decays at least exponentially fast with rate H∞(PX) := − logmaxx PX(x) as the dimension n of X goes to infinity; if PX is absolutely continuous, then only one-dimensional X is sufficient to simulate Y exactly or arbitrarily well. For continuous but not absolutely continuous PX , 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). For the former case, the approximation error for the universal simulation also decays at least exponentially fast; and for the latter case, using a universal simulator, an absolutely continuous random element respect to some continuous distribution (which is known a prior) is also sufficient to simulate another arbitrary random element arbitrarily well.