Achievable key rates for universal simulation of random data with respect to a set of statistical tests

We consider the problem of universal simulation of an unknown source from a certain parametric family of discrete memoryless sources, given a training vector X from that source and given a limited budget of purely random key bits. The goal is to generate a sequence of random vectors {Y_i}, all of the same dimension and the same probability law as the given training vector X, such that a certain, prescribed set of M statistical tests will be satisfied. In particular, for each statistical test, it is required that for a certain event, ε_ℓ, 1 ≤ ℓ ≤ M, the relative frequency ¹/_N Σ_i=1^N 1_εℓ(Y_i) (1_ε(·) being the indicator function of an event ε), would converge, as N → ∞, to a random variable (depending on X), that is typically as close as possible to the expectation of 1_εℓ, (X) with respect to the true unknown source, namely, to the probability of the event ε_ℓ. We characterize the minimum key rate needed for this purpose and demonstrate how this minimum can be approached in principle.