Thinking about the Renyi part of this question again today, I realized that there is a simple and elegant way to show the equivalence of knowing the Renyi entropies and knowing the probabilities (in principle) without taking limits. See Ori's comments, also.

Suppose we have just a finite number outcomes. Then we can place all of the probabilities for each outcome on the diagonal of a large matrix. The Renyi entropies are basically just the traces of the powers of this matrix for integer values of $\alpha$. We would like to show that knowing these trace powers is equivalent to knowing the probabilities themselves. Intuitively, this seems clear, since it is just an overdetermined system of polynomial equations, but a priori it isn't clear that there isn't some weird degeneracy hidden somewhere that would preclude a unique solution. So, we have the trace powers, and as a function of the probabilities, these are just the power sums. We can use the Newton-Girard identities to transform these into the elementary symmetric polynomials. Then we can express the characteristic polynomial of our large matrix as a sum over these. The roots of this polynomial are of course the eigenvalues, which are just the probabilities in question.

Thinking about the Renyi part of this question again today, I realized that there is a simple and elegant way to show the equivalence of knowing the Renyi entropies and knowing the probabilities (in principle) without taking limits. See Ori's comments, also.

Suppose we have just a finite number outcomes. Then we can place all of the probabilities for each outcome on the diagonal of a large matrix. The Renyi entropies are basically just the traces of the powers of this matrix for integer values of $\alpha$. We would like to show that knowing these trace powers is equivalent to knowing the probabilities themselves. Intuitively, this seems clear, since it is just an overdetermined system of polynomial equations, but a priori it isn't clear that there isn't some weird degeneracy hidden somewhere that would preclude a unique solution. So, we have the trace powers, and as a function of the probabilities, these are just the power sums. We can use the Newton-Girard identities to transform these into the elementary symmetric polynomials. Then we can express the characteristic polynomial of our large matrix as a sum over these. The roots of this polynomial are of course the eigenvalues, which are just the probabilities in question.