Efficiency of the empirical distribution for ergodic diffusion (Q1380402)

We consider the problem of estimation of a one-dimensional distribution function \(F(x)\) by the observation of a diffusion process \(\{X_t,0 \leq t\leq T\}\) as \(T\to\infty\). We suppose that the process \(X_t\), \(t\geq 0\), possesses ergodic properties with invariant measure \({\mathbf P}^*\) and \(F(x)= {\mathbf P}^*((-\infty,x])\). We introduce a lower (minimax) bound on the risks of all estimators, then we define the asymptotically efficient estimators as estimators attaining this bound, and finally we show that the empirical distribution function is asymptotically efficient in this problem.