Estimation for the invariant law of an ergodic diffusion process based on high-frequency data

Let a one-dimensional ergodic diffusion process X be observed at time points 0 = t ⁿ ₀ < t ⁿ ₁ < ... < t ⁿ _n such that t ⁿ _n →∞ and n Δ ^1+p _n → 0, where Δ _n = max _1≤i≤n; {pipe}t ⁿ _i - t ⁿ _i-1{pipe}, with p ∈ (0, 1) being a constant depending also on some conditions on X. We consider the nonparametric estimation problems for the invariant distribution and the invariant density. In both problems, we propose some estimators which are asymptotically normal and asymptotically efficient in some functional senses.