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

In this paper, we are concerned with centered Markov Additive Processes {(Xt,Yt)}t∈T where the driving Markov process {Xt}t∈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−1/2Yt given X0. 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.