Uniform law of large numbers and consistency of estimators for Harris diffusions

Consider a family of local martingales depending on a parameter θ running through some compact in Rd. We show that if their quadratic variations are Hölder in θ, then the family satisfies a uniform law of large numbers. We apply it to deduce the almost sure consistency of maximum likelihood estimators for drift parameters of a multidimensional Harris recurrent diffusion, thereby extending a recent result of J.H. van Zanten for one-dimensional ergodic diffusions.