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

Motivated by the Central Limit Theorem, in this paper, we study both universal and non-universal simulations of random variables with an arbitrary target distribution $Q_{Y}$ by general mappings, not limited to linear ones (as in the Central Limit Theorem). 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. Furthermore, we also generalize these results to simulation from Markov processes, and simulation of random elements (or general random variables).