Hostname: page-component-78c5997874-fbnjt Total loading time: 0 Render date: 2024-11-10T14:53:43.117Z Has data issue: false hasContentIssue false
  • English
  • Français

Yaglom limits can depend on the starting state

Part of: Markov processes

Published online by Cambridge University Press:  20 March 2018

R. D. Foley  and
D. R. McDonald
R. D. Foley*
Affiliation:
Georgia Institute of Technology
D. R. McDonald*
Affiliation:
The University of Ottawa
*
* Postal address: Department of Industrial & Systems Engineering, Georgia Institute of Technology, Atlanta, GA 30332-0205, USA. Email address: rfoley@gatech.edu
** Postal address: Department of Mathematics and Statistics, The University of Ottawa, Ottawa, Ontario, K1N 6N5, Canada. Email address: dmdsg@uottawa.ac
Get access Rights & Permissions [Opens in a new window]

Abstract

We construct a simple example, surely known to Harry Kesten, of an R-transient Markov chain on a countable state space S ∪ {δ}, where δ is absorbing. The transition matrix K on S is irreducible and strictly substochastic. We determine the Yaglom limit, that is, the limiting conditional behavior given nonabsorption. Each starting state x ∈ S results in a different Yaglom limit. Each Yaglom limit is an R-1-invariant quasi-stationary distribution, where R is the convergence parameter of K. Yaglom limits that depend on the starting state are related to a nontrivial R-1-Martin boundary.

Keywords

Quasi-stationaryR-transientρ-Martin boundaryYaglom limitgambler's ruineigenvalueeigenvectortime reversalDoob's h-transformationchange of measureinvariant measureharmonic functionsubstochastic

MSC classification

Primary: 60J10: Markov chains (discrete-time Markov processes on discrete state spaces)
Secondary: 60J50: Boundary theory
Type
Original Article
Information
Advances in Applied Probability , Volume 50 , Issue 1 , March 2018 , pp. 1 - 34
Copyright
Copyright © Applied Probability Trust 2018 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1] Alili, L. and Doney, R. A. (2001). Martin boundaries associated with a killed random walk. Ann. Inst. H. Poincaré Prob. Statist. 37, 313–338. Google Scholar
[2] Billingsley, P. (1995). Probability and Measure, 3rd edn. John Wiley, New York. Google Scholar
[3] Breyer, L. A. (1998). Quasistationarity and Martin boundaries: conditioned processes. Preprint. Available at http://www.lbreyer.com/preprints.html. Google Scholar
[4] Clark, P. L. (2012). Sequences and series: a sourcebook. Preprint. Available at http://math.uga.edu/~pete/3100supp.pdf. Google Scholar
[5] Clark, P. L. (2014). Honors calculus. Preprint. Available at http://math.uga.edu/~pete/2400full.pdf. Google Scholar
[6] Doney, R. A. (1998). The Martin boundary and ratio limit theorems for killed random walks. J. London Math. Soc. (2) 58, 761–768. Google Scholar
[7] Doob, J. L. (1959). Discrete potential theory and boundaries. J. Math. Mech. 8, 433–458, 993. Google Scholar
[8] Dynkin, E. B. (1969). Boundary theory of Markov processes (the discrete case). Russian Math. Surveys 24, 42 pp. Google Scholar
[9] Ferrari, P. A. and Rolla, L. T. (2015). Yaglom limit via Holley inequality. Braz. J. Prob. Statist. 29, 413–426. Google Scholar
[10] Foley, R. D. and McDonald, D. R. (2017). Yaglom limits for R-recurrent chains. Available at https://arxiv.org/abs/1709.06610. Google Scholar
[11] Foley, R. D. and McDonald, D. R. (2017). Yaglom limits for R-transient chains and the space-time Martin boundary. Unpublished manuscript. Google Scholar
[12] Hunt, G. A. (1960). Markoff chains and Martin boundaries. Illinois J. Math. 4, 313–340. Google Scholar
[13] Ignatiouk-Robert, I. (2008). Martin boundary of a killed random walk on a half-space. J. Theoret. Prob. 21, 35–68. Google Scholar
[14] Ignatiouk-Robert, I. and Loree, C. (2010). Martin boundary of a killed random walk on a quadrant. Ann. Prob. 38, 1106–1142. Google Scholar
[15] Jacka, S. D. and Roberts, G. O. (1995). Weak convergence of conditioned processes on a countable state space. J. Appl. Prob. 32, 902–916. Google Scholar
[16] Kelly, F. P. (1979). Reversibility and Stochastic Networks. John Wiley, Chichester. Google Scholar
[17] Kemeny, J. G., Snell, J. L. and Knapp, A. W. (1976). Denumerable Markov Chains, 2nd edn. Springer, New York. CrossRefGoogle Scholar
[18] Kesten, H. (1995). A ratio limit theorem for (sub) Markov chains on {1, 2, . . .} with bounded jumps. Adv. App. Prob. 27, 652–691. Google Scholar
[19] Lalley, S. P. (1991). Saddle-point approximations and space-time Martin boundary for nearest-neighbor random walk on a homogeneous tree. J. Theoret. Prob. 4, 701–723. Google Scholar
[20] Lecouvey, C. and Raschel, K. (2015). t-Martin boundary of killed random walks in the quadrant. Available at https://arxiv.org/abs/1509.04193. Google Scholar
[21] Maillard, P (2018). The λ-invariant measures of subcritical Bienaymé–Galton–Watson processes. Bernoulli 24, 297–315. CrossRefGoogle Scholar
[22] Odlyzko, A. M. (1995). Asymptotic enumeration methods. In Handbook of Combinatorics, Elsevier, Amsterdam, pp. 1063–1229. Google Scholar
[23] Pollett, P. K. (1988). Reversibility, invariance and μ-invariance. Adv. Appl. Prob. 20, 600–621. Google Scholar
[24] Pollett, P. K. (1989). The generalized Kolmogorov criterion. Stoch. Process. Appl. 33, 29–44. CrossRefGoogle Scholar
[25] Raschel, K. (2009). Random walks in the quarter plane absorbed at the boundary: exact and asymptotic. Preprint. Available at https://arxiv.org/abs/0902.2785. Google Scholar
[26] Seneta, E. (2006). Non-Negative Matrices and Markov Chains. Springer, New York. Google Scholar
[27] Seneta, E. and Vere-Jones, D. (1966). On quasi-stationary distributions in discrete-time Markov chains with a denumerable infinity of states. J. Appl. Prob. 3, 403–434. Google Scholar
[28] Van Doorn, E. A. (1991). Quasi-stationary distributions and convergence to quasi-stationarity of birth-death processes. Adv. Appl. Prob. 23, 683–700. Google Scholar
[29] Van Doorn, E. A. and Pollett, P. K. (2013). Quasi-stationary distributions for discrete-state models. Europ. J. Operat. Res. 230, 1–14. Google Scholar
[30] Van Doorn, E. A. and Schrijner, P. (1995). Geometric ergodicity and quasi-stationarity in discrete-time birth-death processes. ANZIAM J. 37, 121–144. Google Scholar
[31] Vere-Jones, D. (1967). Ergodic properties of nonnegative matrices. I. Pacific J. Math. 22, 361–386. CrossRefGoogle Scholar
[32] Villemonais, D. (2015). Minimal quasi-stationary distribution approximation for a birth and death process. Electron. J. Prob. 20, 30. Google Scholar
[33] Woess, W. (2000). Random Walks on Infinite Graphs and Groups (Camb. Tracts Math. 138). Cambridge University Press. Google Scholar
[34] Woess, W. (2009). Denumerable Markov Chains: Generating Functions, Boundary Theory, Random Walks on Trees. European Mathematical Society, Zürich. Google Scholar