Deterministic equivalents of additive functionals of recurrent diffusions and drift estimation

Let X _t be a one-dimensional Harris recurrent diffusion, with a drift depending on an unknown parameter θ belonging to some metric compact Θ. We firstly show that all integrable additive functionals of X _t are asymptotically equivalent in probability to some deterministic process v _t . Then we use this result to study the behavior of the maximum likelihood estimator for the parameter θ. Under mild regularity assumptions, we find an upper rate of its convergence as a function of v _t , extending some recent results for ergodic diffusions.