Limit theorems for stationary Markov processes with L2-spectral gap

Déborah Ferré; Loïc Hervé; James Ledoux

Annales de l'I.H.P. Probabilités et statistiques (2012)

  • Volume: 48, Issue: 2, page 396-423
  • ISSN: 0246-0203

Abstract

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Let ( X t , Y t ) t ∈ 𝕋 be a discrete or continuous-time Markov process with state space 𝕏 × ℝ d where 𝕏 is an arbitrary measurable set. Its transition semigroup is assumed to be additive with respect to the second component, i.e. ( X t , Y t ) t ∈ 𝕋 is assumed to be a Markov additive process. In particular, this implies that the first component ( X t ) t ∈ 𝕋 is also a Markov process. Markov random walks or additive functionals of a Markov process are special instances of Markov additive processes. In this paper, the process ( Y t ) t ∈ 𝕋 is shown to satisfy the following classical limit theorems: (a) the central limit theorem, (b) the local limit theorem, (c) the one-dimensional Berry–Esseen theorem, (d) the one-dimensional first-order Edgeworth expansion, provided that we have sup t ∈ ( 0 , 1 ] ∩ 𝕋 𝔼 π , 0 [ | Y t | α ] < ∞ with the expected orderα with respect to the independent case (up to some εgt; 0 for (c) and (d)). For the statements (b) and (d), a Markov nonlattice condition is also assumed as in the independent case. All the results are derived under the assumption that the Markov process ( X t ) t ∈ 𝕋 has an invariant probability distributionπ, is stationary and has the 𝕃 2 ( π ) -spectral gap property (that is, (Xt)t∈ℕ is ρ-mixing in the discrete-time case). The case where ( X t ) t ∈ 𝕋 is non-stationary is briefly discussed. As an application, we derive a Berry–Esseen bound for theM-estimators associated with ρ-mixing Markov chains.

How to cite

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Ferré, Déborah, Hervé, Loïc, and Ledoux, James. "Limit theorems for stationary Markov processes with L2-spectral gap." Annales de l'I.H.P. Probabilités et statistiques 48.2 (2012): 396-423. <http://eudml.org/doc/272051>.

@article{Ferré2012,
abstract = {Let $(X_\{t\},Y_\{t\})_\{t\in \mathbb \{T\}\}$ be a discrete or continuous-time Markov process with state space $\mathbb \{X\}\times \mathbb \{R\}^\{d\}$ where $\mathbb \{X\}$ is an arbitrary measurable set. Its transition semigroup is assumed to be additive with respect to the second component, i.e. $(X_\{t\},Y_\{t\})_\{t\in \mathbb \{T\}\}$ is assumed to be a Markov additive process. In particular, this implies that the first component $(X_\{t\})_\{t\in \mathbb \{T\}\}$ is also a Markov process. Markov random walks or additive functionals of a Markov process are special instances of Markov additive processes. In this paper, the process $(Y_\{t\})_\{t\in \mathbb \{T\}\}$ is shown to satisfy the following classical limit theorems: (a) the central limit theorem, (b) the local limit theorem, (c) the one-dimensional Berry–Esseen theorem, (d) the one-dimensional first-order Edgeworth expansion, provided that we have $\sup _\{t\in (0,1]\cap \mathbb \{T\}\}\mathbb \{E\}_\{\pi ,0\}[|Y_\{t\}|^\{\alpha \}]<\infty $ with the expected orderα with respect to the independent case (up to some εgt; 0 for (c) and (d)). For the statements (b) and (d), a Markov nonlattice condition is also assumed as in the independent case. All the results are derived under the assumption that the Markov process $(X_\{t\})_\{t\in \mathbb \{T\}\}$ has an invariant probability distributionπ, is stationary and has the $\mathbb \{L\}^\{2\}(\pi )$ -spectral gap property (that is, (Xt)t∈ℕ is ρ-mixing in the discrete-time case). The case where $(X_\{t\})_\{t\in \mathbb \{T\}\}$ is non-stationary is briefly discussed. As an application, we derive a Berry–Esseen bound for theM-estimators associated with ρ-mixing Markov chains.},
author = {Ferré, Déborah, Hervé, Loïc, Ledoux, James},
journal = {Annales de l'I.H.P. Probabilités et statistiques},
keywords = {Markov additive process; central limit theorems; Berry–Esseen bound; edgeworth expansion; spectral method; ρ-mixing; M-estimator; discrete or continuous-time Markov process; Markov random walk; additive functional; stationary Markov process; -spectral gap property; geometric ergodicity; uniform ergodicity; -mixing Markov chain; Fourier operator; central limit theorem; functional central limit theorem; local limit theorem; Berry-Esseen theorem; Edgeworth expansion; -estimator},
language = {eng},
number = {2},
pages = {396-423},
publisher = {Gauthier-Villars},
title = {Limit theorems for stationary Markov processes with L2-spectral gap},
url = {http://eudml.org/doc/272051},
volume = {48},
year = {2012},
}

TY - JOUR
AU - Ferré, Déborah
AU - Hervé, Loïc
AU - Ledoux, James
TI - Limit theorems for stationary Markov processes with L2-spectral gap
JO - Annales de l'I.H.P. Probabilités et statistiques
PY - 2012
PB - Gauthier-Villars
VL - 48
IS - 2
SP - 396
EP - 423
AB - Let $(X_{t},Y_{t})_{t\in \mathbb {T}}$ be a discrete or continuous-time Markov process with state space $\mathbb {X}\times \mathbb {R}^{d}$ where $\mathbb {X}$ is an arbitrary measurable set. Its transition semigroup is assumed to be additive with respect to the second component, i.e. $(X_{t},Y_{t})_{t\in \mathbb {T}}$ is assumed to be a Markov additive process. In particular, this implies that the first component $(X_{t})_{t\in \mathbb {T}}$ is also a Markov process. Markov random walks or additive functionals of a Markov process are special instances of Markov additive processes. In this paper, the process $(Y_{t})_{t\in \mathbb {T}}$ is shown to satisfy the following classical limit theorems: (a) the central limit theorem, (b) the local limit theorem, (c) the one-dimensional Berry–Esseen theorem, (d) the one-dimensional first-order Edgeworth expansion, provided that we have $\sup _{t\in (0,1]\cap \mathbb {T}}\mathbb {E}_{\pi ,0}[|Y_{t}|^{\alpha }]<\infty $ with the expected orderα with respect to the independent case (up to some εgt; 0 for (c) and (d)). For the statements (b) and (d), a Markov nonlattice condition is also assumed as in the independent case. All the results are derived under the assumption that the Markov process $(X_{t})_{t\in \mathbb {T}}$ has an invariant probability distributionπ, is stationary and has the $\mathbb {L}^{2}(\pi )$ -spectral gap property (that is, (Xt)t∈ℕ is ρ-mixing in the discrete-time case). The case where $(X_{t})_{t\in \mathbb {T}}$ is non-stationary is briefly discussed. As an application, we derive a Berry–Esseen bound for theM-estimators associated with ρ-mixing Markov chains.
LA - eng
KW - Markov additive process; central limit theorems; Berry–Esseen bound; edgeworth expansion; spectral method; ρ-mixing; M-estimator; discrete or continuous-time Markov process; Markov random walk; additive functional; stationary Markov process; -spectral gap property; geometric ergodicity; uniform ergodicity; -mixing Markov chain; Fourier operator; central limit theorem; functional central limit theorem; local limit theorem; Berry-Esseen theorem; Edgeworth expansion; -estimator
UR - http://eudml.org/doc/272051
ER -

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