  A quantity of interest to characterise continuous-valued stochastic processes
is the differential entropy rate. The rate of convergence of many properties of
LRD processes is slower than might be expected, based on the intuition for
conventional processes, e.g. Markov processes. Is this also true of the entropy
rate?
  In this paper we consider the properties of the differential entropy rate of
stochastic processes that have an autocorrelation function that decays as a
power law. We show that power law decaying processes with similar
autocorrelation and spectral density functions, Fractional Gaussian Noise and
ARFIMA(0,d,0), have different entropic properties, particularly for negatively
correlated parameterisations. Then we provide an equivalence between the mutual
information between past and future and the differential excess entropy for
stationary Gaussian processes, showing the finiteness of this quantity is the
boundary between long and short range dependence. Finally, we analyse the
convergence of the conditional entropy to the differential entropy rate and
show that for short range dependence that the rate of convergence is of the
order $O(n^{-1})$, but it is slower for long range dependent processes and
depends on the Hurst parameter.
