Kannan, D.; Zhang, Jingxiao Self-interacting Markov chains: Some asymptotics. (English) Zbl 1158.60045 Stochastic Anal. Appl. 27, No. 1, 196-219 (2009). Summary: Guided by the self-interaction mechanisms introduced in [M. Benaim, M. Ledoux and O. Raimond, Probab. Theory Relat. Fields 122, No. 1, 1–41 (2002; Zbl 1042.60060)] and in [P. Del Moral and L. Miclo, Stochastic Anal. Appl. 24, No. 3, 615–660 (2006; Zbl 1093.60068)], we present a more general definition of self-interacting Markov chains (SIMCs) (than in Del Moral and Miclo [loc. cit.] and Benaim et al. [loc. cit.]). We then establish, for particular self-interaction mechanisms, a stability theorem with error estimation, two central limit theorems, two functional central limit theorems, and the large deviation principle. MSC: 60K35 Interacting random processes; statistical mechanics type models; percolation theory 60J10 Markov chains (discrete-time Markov processes on discrete state spaces) 60F10 Large deviations 60F17 Functional limit theorems; invariance principles Keywords:FCLT; LDP; Markov chain; self-interaction Citations:Zbl 1042.60060; Zbl 1093.60068 PDFBibTeX XMLCite \textit{D. Kannan} and \textit{J. Zhang}, Stochastic Anal. Appl. 27, No. 1, 196--219 (2009; Zbl 1158.60045) Full Text: DOI References: [1] DOI: 10.2748/tmj/1178243286 · Zbl 0178.21103 · doi:10.2748/tmj/1178243286 [2] DOI: 10.1007/s004400100161 · Zbl 1042.60060 · doi:10.1007/s004400100161 [3] DOI: 10.1016/S0246-0203(03)00028-1 · Zbl 04577776 · doi:10.1016/S0246-0203(03)00028-1 [4] DOI: 10.1088/0305-4470/6/9/014 · doi:10.1088/0305-4470/6/9/014 [5] DOI: 10.1080/07362990600632029 · Zbl 1093.60068 · doi:10.1080/07362990600632029 [6] Dembo A., Large Deviations Techniques and Applications., 2. ed. (1998) · Zbl 0896.60013 [7] Dobrushin R. L., Problems Inform. Transmission 7 pp 149– (1971) [8] Dobrushin R. L., Problems Infrom. Transmission 7 pp 235– (1971) [9] Durrett R., Probability: Theory and Examples (2005) · Zbl 1202.60002 [10] Hall P., Martingale Limit Theory and Its Applications (1980) · Zbl 0462.60045 [11] Liggett T. M., Interacting Particle Systems (1985) [12] DOI: 10.1214/074921706000000103 · Zbl 1125.82014 · doi:10.1214/074921706000000103 [13] Sethuraman S., Electronic J. Prob. 10 pp 1221– (2005) [14] DOI: 10.1007/BFb0079128 · doi:10.1007/BFb0079128 [15] DOI: 10.1016/0001-8708(70)90034-4 · Zbl 0312.60060 · doi:10.1016/0001-8708(70)90034-4 [16] Strassen V., Proc. 5th Berkeley Symp. 2 pp 315– (1967) This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.