# Nonparametric estimation of the stationary density and the transition density of a Markov chain

Abstract : In this paper, we study first the problem of nonparametric estimation of the stationary density $f$ of a discrete-time Markov chain $(X_i)$. We consider a collection of projection estimators on finite dimensional linear spaces. We select an estimator among the collection by minimizing a penalized contrast. The same technique enables to estimate the density $g$ of $(X_i, X_{i+1})$ and so to provide an adaptive estimator of the transition density $\pi=g/f$. We give bounds in $L^2$ norm for these estimators and we show that they are adaptive in the minimax sense over a large class of Besov spaces. Some examples and simulations are also provided.
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https://hal.archives-ouvertes.fr/hal-00115457
Contributor : Claire Lacour <>
Submitted on : Wednesday, January 9, 2008 - 12:06:04 PM
Last modification on : Friday, September 20, 2019 - 4:34:02 PM
Long-term archiving on: Tuesday, September 21, 2010 - 3:08:05 PM

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### Citation

Claire Lacour. Nonparametric estimation of the stationary density and the transition density of a Markov chain. Stochastic Processes and their Applications, Elsevier, 2008, 118 (2), pp 232-260. ⟨10.1016/j.spa.2007.04.013⟩. ⟨hal-00115457v2⟩

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