A local limit theorem for densities of the additive component of a finite Markov Additive Process

Loïc Hervé (1, 2) , James Ledoux (1, 2)

1 IRMAR - Institut de Recherche Mathématique de Rennes
2 INSA Rennes - Institut National des Sciences Appliquées - Rennes

In this paper, we are concerned with centered Markov Additive Processes $\{(X_t,Y_t)\}_{t\in\mathbb{T}}$ where the driving Markov process $\{X_t\}_{t\in\mathbb{T}}$ has a finite state space. Under suitable conditions, we provide a local limit theorem for the density of the absolutely continuous part of the probability distribution of $t^{-1/2}Y_t$ given $X_0$. The rate of convergence and the moment condition are the expected ones with respect to the i.i.d case. An application to the joint distribution of local times of a finite jump process is sketched.

Mots clés: Gaussian approximation; Local time; Spectral method; Markov random walk

Domaines: Probabilités [math.PR]; Statistiques [math.ST]; Théorie [stat.TH]

Loïc Hervé, James Ledoux. A local limit theorem for densities of the additive component of a finite Markov Additive Process. Statistics and Probability Letters, 2013, 83 (9), pp.2119-2128. ⟨10.1016/j.spl.2013.05.032⟩. ⟨hal-00837498⟩