Polynomial deviation bounds for recurrent Harris processes having general state space
Consider a strong Markov process in continuous time, taking values in some Polish state space. Recently, Douc
Penalized nonparametric drift estimation for a continuously observed one-dimensional diffusion process
Let be a one dimensional positive recurrent diffusion continuously observed on [0,] . We consider a non parametric estimator of the drift function on a given interval. Our estimator, obtained using a penalized least square approach, belongs to a finite dimensional functional space, whose dimension is selected according to the data. The non-asymptotic risk-bound reaches the minimax optimal rate of convergence when → ∞. The main point of our work is that we do not suppose the process to be in stationary...
Polynomial bounds in the Ergodic theorem for one-dimensional diffusions and integrability of hitting times
Let be a one-dimensional positive recurrent diffusion with initial distribution and invariant probability . Suppose that for some >1, ∈ℝ such that ∀∈ℝ, and , where is the hitting time of . For such a diffusion, we derive non-asymptotic deviation bounds of the form ℙ(|(1/)0 ( ) d−()|≥)≤()(1/ /2)(1/ )(). Here bounded or bounded and compactly supported and ()=‖‖∞ when is bounded and ()=(||) when is bounded and...
Spectral gaps and exponential integrability of hitting times for linear diffusions
Let be a regular continuous positively recurrent Markov process with state space ℝ, scale function and speed measure . For ∈ℝ denote +=sup≥ (], +∞[)(()−()), −=sup≤ (]−∞; [)(()−()). It is well known that the finiteness of ± is equivalent to the existence of spectral gaps of generators associated with . We show how these quantities appear independently in the study of the exponential moments of hitting times of . Then...
Spectral condition, hitting times and Nash inequality
Let be a -symmetric Hunt process on a LCCB space . For an open set , let be the exit time of from and be the generator of the process killed when it leaves . Let and . We give necessary and sufficient conditions for in terms of the behavior near the origin of the spectral measure of . When , , by means of this condition we derive the Nash inequality for the killed process. In the diffusion case this permits to show that the existence of moments of order for implies the...
Penalized nonparametric drift estimation for a continuously observed one-dimensional diffusion process
Let be a one dimensional positive recurrent diffusion continuously observed on [0,] . We consider a non parametric estimator of the drift function on a given interval. Our estimator, obtained using a penalized least square approach, belongs to a finite dimensional functional space, whose dimension is selected according to the data. The non-asymptotic risk-bound reaches the minimax optimal rate of convergence when → ∞. The main point of our work is that we do not suppose the process to be in stationary...
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