LAMN in a class of parametric models for null recurrent diffusions

We study statistical models for one-dimensional diffusions which are null recurrent. A first parameter in the drift is the principal one, and determines regular varying rates of convergence for the score and the information process. A finite number of other parameters, of secondary importance, introduces additional flexibility for the modelization of the drift, and does not perturb the null recurrent behaviour. Under time-continuous observation we obtain local asymptotic mixed normality, state a local asymptotic minimax bound, and specify asymptotically optimal estimators.