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