  The purpose of this note is to prove a lower bound for the estimation of the
memory parameter of a stationary long memory process. The memory parameter is
defined here as the index of regular variation of the spectral density at 0.
The rates of convergence obtained in the literature assume second order regular
variation of the spectral density at zero. In this note, we do not make this
assumption, and show that the rates of convergence in this case can be
extremely slow. We prove that the log-periodogram regression (GPH) estimator
achieves the optimal rate of convergence for Gaussian long memory processes.
