Luís Ramos

When a diffusion is ergodic its transition density converges to its invariant density, see Durrett (1998). This convergence enabled us to introduce a sample partitioning technique that gives in each sub-sample, maximum likelihood estimators. The averages of these being a natural choice as estimators. To compare our estimators with the optimal we obtained from martingale estimating functions, see Sorensen (1998), we used the Ornstein-Uhlenbeck process for which exact simulations can be carried out.

Some asymptotic expansions non necessarily related to the central limit theorem are discussed. After observing that the smoothing inequality of Esseen implies the proximity, in the Kolmogorov distance sense, of the distributions of the random variables of two random sequences satisfying a sort of general asymptotic relation, two instances of this observation are presented. A first example, partially motivated by the the statistical theory of high precision measurements, is given by a uniform asymptotic approximation to $(g(X+ μ_n))_{n ın \mathbbm{N}}$, where $g$ is some smooth function, $X$ is a random variable having a moment and a bounded density and $(μ_{n})_{n ın \mathbbm{N}}$ is a sequence going to infinity; the multivariate case as well as the proofs and a complete set of references will be published elsewhere. We next present a second class of examples given by a randomization of the interesting parameter in some classical asymptotic formulas, namely, a generic Laplace's type integral, by the sequence $(μ_n X)_{n ın \mathbbm{N}}$, $X$ being a Gamma distributed random variable. Finally, a simulation study of this last example is presented in order to stress the quality of asymptotic approximations proposed.