Limit theorems for stationary Markov processes with $L^2$-spectral gap

Ferré, Déborah; Hervé, Loïc; Ledoux, James

Annales de l'I.H.P. Probabilités et statistiques, Tome 48 (2012) no. 2, pp. 396-423.

Soit ${(X_t,Y_t)}_{t\in 𝕋}$ un processus de Markov en temps discret ou continu et d’espace d’état $𝕏\times {ℝ}^d$ où $𝕏$ est un ensemble mesurable quelconque. Son semi-groupe de transition est supposé additif suivant la seconde composante, i.e. ${(X_t,Y_t)}_{t\in 𝕋}$ est un processus additif Markovien. En particulier, ceci implique que la première composante ${\left(X_t\right)}_{t\in 𝕋}$ est également un processus de Markov. Les marches aléatoires Markoviennes ou les fonctionnelles additives d’un processus de Markov sont des exemples de processus additifs Markoviens. Dans cet article, on montre que le processus ${\left(Y_t\right)}_{t\in 𝕋}$ satisfait les théorèmes limites classiques suivants : (a) le théorème de la limite centrale, (b) le théorème limite local, (c) le théorème uniforme de Berry-Esseen en dimension un, (d) le développement d’Edgeworth d’ordre un en dimension un, pourvu que la condition de moment ${sup}_{t\in (0,1]\cap 𝕋}{𝔼}_{\pi ,0}\left[\right|Y_t{|}^{\alpha }]<\infty$ soit satisfaite, avec l’ordre attendu α du cas indépendant (à un ε > 0 près pour (c) et (d)). Pour les énoncés (b) et (d), il faut ajouter une condition nonlattice comme dans le cas indépendant. Tous les résultats sont obtenus sous l’hypothèse d’un processus de Markov ${\left(X_t\right)}_{t\in 𝕋}$ admettant une mesure de probabilité invariante π et possédant la propriété de trou spectral sur ${𝕃}^2\left(\pi \right)$ (c’est à dire, (Xt)t∈ℕ est ρ-mélangeante dans le cas du temps discret). Le cas où ${\left(X_t\right)}_{t\in 𝕋}$ est non-stationnaire est brièvement abordé. Nous appliquons nos résultats pour obtenir une borne de Berry-Esseen pour les M-estimateurs associés aux chaînes de Markov ρ-mélangeantes.

Let ${(X_t,Y_t)}_{t\in 𝕋}$ be a discrete or continuous-time Markov process with state space $𝕏\times {ℝ}^d$ where $𝕏$ is an arbitrary measurable set. Its transition semigroup is assumed to be additive with respect to the second component, i.e. ${(X_t,Y_t)}_{t\in 𝕋}$ is assumed to be a Markov additive process. In particular, this implies that the first component ${\left(X_t\right)}_{t\in 𝕋}$ 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 ${\left(Y_t\right)}_{t\in 𝕋}$ 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 ${sup}_{t\in (0,1]\cap 𝕋}{𝔼}_{\pi ,0}\left[\right|Y_t{|}^{\alpha }]<\infty$ with the expected order α with respect to the independent case (up to some ε > 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 ${\left(X_t\right)}_{t\in 𝕋}$ has an invariant probability distribution π, is stationary and has the ${𝕃}^2\left(\pi \right)$ -spectral gap property (that is, (Xt)t∈ℕ is ρ-mixing in the discrete-time case). The case where ${\left(X_t\right)}_{t\in 𝕋}$ is non-stationary is briefly discussed. As an application, we derive a Berry-Esseen bound for the M-estimators associated with ρ-mixing Markov chains.

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