  Determining entropy rates of stochastic processes is a fundamental and
difficult problem, with closed-form solutions known only for specific cases.
This paper pushes the state-of-the-art by solving the problem for Hidden Markov
Models (HMMs) and Renyi entropies.
  While the problem for Markov chains reduces to studying the growth of a
matrix product, computations for HMMs involve \emph{products of random
matrices}. As a result, this case is much harder and no explicit formulas have
been known so far. We show how to circumvent this issue for Renyi entropy of
integer orders, reducing the problem again to a \emph{single matrix products}
where the matrix is formed from transition and emission probabilities by means
of tensor product.
  To obtain results in the asymptotic setting, we use a novel technique for
determining the growth of non-negative matrix powers. The classical approach is
the Frobenius-Perron theory, but it requires positivity assumptions; we instead
work directly with the spectral formula. As a consequence, our results do not
suffer from limitations such as irreducibility and aperiodicity. This improves
our understanding of the entropy rate even for standard (unhidden) Markov
chains.
  A recently published side-channel attack against RSA was proven effective
using our result, specialized to order 2.
