Höpfner, R.; Löcherbach, E. Limit theorems for null recurrent Markov processes. (English) Zbl 1018.60074 Mem. Am. Math. Soc. 768, 92 p. (2003). The aim of the paper is to give a self-contained treatment of weak convergence of martingales and integrable additive functionals in general Harris recurrent Markov processes in continuous time. The paper provides a new look on a partially very old topic. The authors use the notation of the book by J. Jacod and A. N. Shiryaev [“Limit theorems for stochastic processes” (1987; Zbl 0635.60021)].The paper is organized as follows. In the introduction the preliminary versions of the main results are given. The first two sections also have an introductory character: Section 1 deals with Harris recurrence, and Section 2 with stable processes and classical convergence to stable laws. The main results of the paper are formulated in Section 3: Subsection 3.1 is devoted to processes which admit a recurrent atom and thus i.i.d. life cycles; Subsection 3.2 gives a family of examples which apply the result ‘with life cycles’ to classical one-dimensional diffusions, with strong reference to Khasminskij (his explicit representation of tails of life-cycle length distributions in null recurrent one-dimensional diffusions is a key tool there); Subsection 3.3 states the general result (without assuming existence of life cycles) under minimal hypotheses. The proofs of these results are contained in Sections 4-7. The paper contributes to make existing theorems on weak convergence of martingales and integrable additive functionals in null recurrent Markov processes better known in the probabilistic and statistical community, and is useful as a self-contained reference in applications such as statistical inference for stochastic processes. Reviewer: Pavel Gapeev (Moskva) Cited in 24 Documents MSC: 60J25 Continuous-time Markov processes on general state spaces 60F17 Functional limit theorems; invariance principles 60G44 Martingales with continuous parameter Keywords:Markov processes in continuous time; Harris recurrence; weak convergence of martingales; integrable additive functionals Citations:Zbl 0635.60021 PDFBibTeX XMLCite \textit{R. Höpfner} and \textit{E. Löcherbach}, Limit theorems for null recurrent Markov processes. Providence, RI: American Mathematical Society (AMS) (2003; Zbl 1018.60074) Full Text: DOI