R-positivity, quasi-stationary distributions and ratio limit theorems for a class of probabilistic automata

Annals of Applied Probability

Volumn 6, Issue 2, 1996, Pages 577-616

  Ferrari, P A ^a   Kesten, H ^b   Martinez S ^c  

^a UNIVERSITY OF SÃO PAULO   (Brazil)

^b Cornell University ^*  (United States)

^c UNIVERSITY OF CHILE   (Chile)

Author keywords

Absorbing Markov chain; Central limit theorem; Quasi stationary distribution; R positivity; Ratio limit theorem; Yaglom limit

Indexed keywords

EID: 0030502869     PISSN: 10505164     EISSN: None     Source Type: Journal    
DOI: 10.1214/aoap/1034968146     Document Type: Article

References (14)

1. AIZENMAN, M. and BARSKY, D. J. (1987). Sharpness of the phase transition in percolation models. Comm. Math. Phys. 108 489-526.
2. BEZUIDENHOUT, C. and GRIMMETT, G. (1991). Exponential decay for subcritical contact and percolation processes. Ann. Probab. 19 984-1009.
3. CAMPANINO, M., CHAYES, J. T. and CHAYES, L. (1991). Gaussian fluctuations of connectivities in the subcritical regime of percolation. Probab. Theory Related Fields 88 269-341.
4. CHUNG, K. L. (1967). Markov Chains with Stationary Transition Probabilities, 2nd ed. Springer, New York.
5. DURRETT, R. (1988). Lecture Notes on Particle Systems and Percolation. Wadsworth and Brooks/Cole, Belmont, CA.
6. FERRARI, P. A., KESTEN, H., MARTÍNEZ, S. and PICCO, P. (1995). Existence of quasi-stationary distributions. A renewal dynamical approach. Ann. Probab. 23 501-521.
7. GRIMMETT, G. (1989). Percolation. Springer, New York.
8. KESTEN, H. (1995). A ratio limit theorem for (sub) Markov chains on {1,2,...} with bounded jumps. Adv. in Appl. Probab. 27 652-691.
9. NGUYEN, B. G. and YANG, W-S. (1995). Gaussian limit for critical oriented percolation in high dimensions. J. Statist. Phys. 78 841-876.
10. PAKES, A. (1995). Quasi-stationary laws for Markov processes: examples of an always proximate absorbing state. Adv. in Appl. Probab. 27 120-145.
11. ROBERTS, G. O. and JACKA, S. D. (1994). Weak convergence of conditioned birth and death processes. J. Appl. Probab. 31 90-100.
12. SENETA, E. and VERE-JONES, D. (1966). On quasi-stationary distributions in discrete-time Markov chains with a denumerable infinity of states. J. Appl. Probab. 3 403-434.
13. VERE-JONES, D. (1967). Ergodic properties of nonnegative matrices I. Pacific J. Math. 22 361-396.
14. YAGLOM, A. M. (1947). Certain limit theorems of the theory of branching stochastic processes. Dokl. Akad. Nauk SSSR 56 797-798.