Yaglom limit via Holley inequality. (English) Zbl 1321.60142

Summary: Let \(S\) be a countable set provided with a partial order and a minimal element. Consider a Markov chain on \(S\cup\{0\}\) absorbed at \(0\) with a quasi-stationary distribution. We use the Holley inequality to obtain sufficient conditions under which the following holds: the trajectory of the chain starting from the minimal state is stochastically dominated by the trajectory of the chain starting from any probability on \(S\), when both are conditioned to nonabsorption until a certain time. Moreover, the Yaglom limit corresponding to this deterministic initial condition is the unique minimal quasi-stationary distribution in the sense of stochastic order. As an application, we provide new proofs to classical results in the field.


60J10 Markov chains (discrete-time Markov processes on discrete state spaces)
60E15 Inequalities; stochastic orderings
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[1] Cavender, J. A. (1978). Quasi-stationary distributions of birth-and-death processes. Adv. in Appl. Probab. 10 , 570-586. · Zbl 0381.60068
[2] Collet, P., Martínez, S. and San Martín, J. (2013). Markov Chains, Diffusions and Dynamical Systems. Probability and Its Applications (New York) . Heidelberg: Springer. · Zbl 1261.60002
[3] Daley, D. J. (1969). Quasi-stationary behaviour of a left-continuous random walk. Ann. Math. Statist. 40 , 532-539. · Zbl 0175.46801
[4] van Doorn, E. A. and Schrijner, P. (1995a). Geometric ergodicity and quasi-stationarity in discrete-time birth-death processes. J. Austral. Math. Soc. Ser. B 37 , 121-144. · Zbl 0856.60080
[5] van Doorn, E. A. and Schrijner, P. (1995b). Ratio limits and limiting conditional distributions for discrete-time birth-death processes. J. Math. Anal. Appl. 190 , 263-284. · Zbl 0821.60088
[6] Ferrari, P. A., Kesten, H., Martinez, S. and Picco, P. (1995). Existence of quasi-stationary distributions. A renewal dynamical approach. Ann. Probab. 23 , 501-521. · Zbl 0827.60061
[7] Ferrari, P. A., Martinez, S. and Picco, P. (1991). Some properties of quasi-stationary distributions in the birth and death chains: A dynamical approach. In Instabilities and Nonequilibrium Structures, III , Valparaíso , 1989. Math. Appl. 64 , 177-187. Dordrecht: Kluwer Acad. Publ.
[8] Ferrari, P. A., Martínez, S. and Picco, P. (1992). Existence of nontrivial quasi-stationary distributions in the birth-death chain. Adv. in Appl. Probab. 24 , 795-813. · Zbl 0769.60063
[9] Georgii, H.-O., Häggström, O. and Maes, C. (2001). The random geometry of equilibrium phases. In Phase Transitions and Critical Phenomena, Vol. 18 . 1-142. San Diego, CA: Academic Press.
[10] Holley, R. (1974). Remarks on the \(\mathrm{FKG}\) inequalities. Comm. Math. Phys. 36 , 227-231.
[11] Iglehart, D. L. (1974). Random walks with negative drift conditioned to stay positive. J. Appl. Probab. 11 , 742-751. · Zbl 0302.60038
[12] Kesten, H. (1995). A ratio limit theorem for (sub) Markov chains on \(\{1,2,\ldots\}\) with bounded jumps. Adv. in Appl. Probab. 27 , 652-691. · Zbl 0829.60059
[13] Lindvall, T. (1999). On Strassen’s theorem on stochastic domination. Electron. Commun. Probab. 4 , 51-59 (electronic). · Zbl 0938.60013
[14] Pollett, P. K. (2014). Quasi-stationary distributions: A bibliography. Available at .
[15] Seneta, E. (1966). Quasi-stationary behaviour in the random walk with continuous time. Austral. J. Statist. 8 , 92-98. · Zbl 0147.16303
[16] 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. · Zbl 0147.36603
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