Displaying similar documents to “Polynomial bounds in the Ergodic theorem for one-dimensional diffusions and integrability of hitting times”

Quasi-compactness and mean ergodicity for Markov kernels acting on weighted supremum normed spaces

Loïc Hervé (2008)

Annales de l'I.H.P. Probabilités et statistiques

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Let be a Markov kernel on a measurable space with countably generated -algebra, let :→[1, +∞[ such that ≤ with ≥0, and let ℬ w be the space of measurable functions on satisfying ‖‖=sup{()|()|, ∈}<+∞. We prove that is quasi-compact on ( ℬ w , ∥ · ∥ w ) if and only if, for all f ∈ ℬ w , ( 1 n ∑ k = 1 n P k f ) n contains a subsequence converging in ℬ w to=∑ () , where the ’s are non-negative bounded measurable functions on and the ’s...

Polynomial deviation bounds for recurrent Harris processes having general state space

Eva Löcherbach, Dasha Loukianova (2013)

ESAIM: Probability and Statistics

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Consider a strong Markov process in continuous time, taking values in some Polish state space. Recently, Douc et al. [Stoc. Proc. Appl. 119, (2009) 897–923] introduced verifiable conditions in terms of a supermartingale property implying an explicit control of modulated moments of hitting times. We show how this control can be translated into a control of polynomial moments of abstract regeneration times which are obtained by using...

Strong law of large numbers for branching diffusions

János Engländer, Simon C. Harris, Andreas E. Kyprianou (2010)

Annales de l'I.H.P. Probabilités et statistiques

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Let be the branching particle diffusion corresponding to the operator +(2−) on ⊆ℝ (where ≥0 and ≢0). Let denote the generalized principal eigenvalue for the operator + on and assume that it is finite. When >0 and +− satisfies certain spectral theoretical conditions, we prove that the random measure {− } converges almost surely in the vague topology as tends to infinity. This result...

Spectral gaps and exponential integrability of hitting times for linear diffusions

Oleg Loukianov, Dasha Loukianova, Shiqi Song (2011)

Annales de l'I.H.P. Probabilités et statistiques

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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...

Limit theorems for some functionals with heavy tails of a discrete time Markov chain

Patrick Cattiaux, Mawaki Manou-Abi (2014)

ESAIM: Probability and Statistics

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Consider an irreducible, aperiodic and positive recurrent discrete time Markov chain ( ≥ 0) with invariant distribution . We shall investigate the long time behaviour of some functionals of the chain, in particular the additive functional S n = ∑ i = 1 n f ( X i ) S n = ∑ i = 1 n f ( X i ) for a possibly non square integrable function. To this end we shall link ergodic properties of the chain to mixing properties, extending known results in the continuous time case. We will then...

Linear diffusion with stationary switching regime

Xavier Guyon, Serge Iovleff, Jian-Feng Yao (2010)

ESAIM: Probability and Statistics

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Let be a Ornstein–Uhlenbeck diffusion governed by a stationary and ergodic process : ddd. We establish that under the condition with the stationary distribution of the regime process , the diffusion is ergodic. We also consider conditions for the existence of moments for the invariant law of when is a Markov jump process having a finite number of states. Using results on random difference equations on one hand and the fact that conditionally to , is Gaussian on the other...

Penalized nonparametric drift estimation for a continuously observed one-dimensional diffusion process

Eva Löcherbach, Dasha Loukianova, Oleg Loukianov (2011)

ESAIM: Probability and Statistics

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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...