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The random variable simulation problem consists in using a k-dimensional i.i.d. random vector Xk with distribution PXk to simulate an n-dimensional i.i.d. random vector Yn so that its distribution is approximately QYn. In contrast to previous works, in this paper we use the Re&#x0301;nyi divergences of all orders to measure the level of approximation. We also characterize the asymptotics of normalized Re&#x0301;nyi divergences as well as the Re&#x0301;nyi conversion rates. The latter are defined as the supremum of n/k such that the Re&#x0301;nyi divergences vanish asymptotically. In addition, when the Re&#x0301;nyi parameter is &#x2208;(0,1), the Re&#x0301;nyi conversion rates equal the ratio of the Shannon entropies H(Px)/H(Qy), which is consistent with traditional results, in which the Re&#x0301;nyi parameter is equal to 1. When the Re&#x0301;nyi parameter is &#x2208; (0, 1), the Re&#x0301;nyi conversion rates are, in general, smaller than H(Px)/H(Qy). In addition to the standard Re&#x0301;nyi divergence, we consider two new divergences - the max-Re&#x0301;nyi divergence D&#x03B1;max(P,Q) and the sum-Re&#x0301;nyi divergence D&#x03B1;+(P, Q), and use them (with order infinity) as approximation measures. These two measures with order infinity are very strong since for any &#x03F5;&gt;0, D&#x221E;max(P,Q)&#x2264;&#x03F5; or D&#x221E;+(P,Q)&#x2264; &#x03F5; implies e-&#x03F5;&#x2264;P(x)/Q(x)&#x2264;e&#x03F5;, &#x2200;x.

Simulation of Random Variables under Rényi Divergence Measures of All Orders