Convergence of integrable additive functionals of Markov processes is an old topic going back to D. A. Darling and M. Kac refTrans. Amer. Math. Soc. 84 (1957), 444--458; MR0084222 (18,832a). The aim of this monograph is to state and prove a general result on weak convergence of martingales and additive functionals in Harris recurrent processes in continuous time. In the ergodic case, the martingale limit theorem applies. In most null recurrent cases, there is no longer convergence in probability of the martingale brackets but only convergence in law and a more intricate proof is needed. For the reader's convenience, the first sections are devoted to the main definitions and properties of Harris recurrent processes, stable increasing processes and Mittag-Leffler processes. These last processes acting as time change for the Brownian motion provide the class of limit processes for the rescaled additive functionals. It is noted that the convergence result may be applied in the context of local asymptotic statistics to convergence of statistical experiments and of maximum likelihood estimators.
Reviewed by Dominique Lộpingle