[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.

[2]
Billingsley, P. (1995). Probability and Measure, 3rd edn. John Wiley, New York.

[6]
Doney, R. A. (1998). The Martin boundary and ratio limit theorems for killed random walks. J. London Math. Soc. (2)
58, 761–768.

[7]
Doob, J. L. (1959). Discrete potential theory and boundaries. J. Math. Mech.
8, 433–458, 993.

[8]
Dynkin, E. B. (1969). Boundary theory of Markov processes (the discrete case). Russian Math. Surveys
24, 42 pp.

[9]
Ferrari, P. A. and Rolla, L. T. (2015). Yaglom limit via Holley inequality. Braz. J. Prob. Statist.
29, 413–426.

[11]
Foley, R. D. and McDonald, D. R. (2017). Yaglom limits for *R*-transient chains and the space-time Martin boundary. Unpublished manuscript.

[12]
Hunt, G. A. (1960). Markoff chains and Martin boundaries. Illinois J. Math.
4, 313–340.

[13]
Ignatiouk-Robert, I. (2008). Martin boundary of a killed random walk on a half-space. J. Theoret. Prob.
21, 35–68.

[14]
Ignatiouk-Robert, I. and Loree, C. (2010). Martin boundary of a killed random walk on a quadrant. Ann. Prob.
38, 1106–1142.

[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.

[16]
Kelly, F. P. (1979). Reversibility and Stochastic Networks. John Wiley, Chichester.

[17]
Kemeny, J. G., Snell, J. L. and Knapp, A. W. (1976). Denumerable Markov Chains, 2nd edn. Springer, New York.

[18]
Kesten, H. (1995). A ratio limit theorem for (sub) Markov chains on {1, 2, . . .} with bounded jumps. Adv. App. Prob.
27, 652–691.

[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.

[21]
Maillard, P (2018). The λ-invariant measures of subcritical Bienaymé–Galton–Watson processes. Bernoulli
24, 297–315.

[22]
Odlyzko, A. M. (1995). Asymptotic enumeration methods. In Handbook of Combinatorics, Elsevier, Amsterdam, pp. 1063–1229.

[23]
Pollett, P. K. (1988). Reversibility, invariance and μ-invariance. Adv. Appl. Prob.
20, 600–621.

[24]
Pollett, P. K. (1989). The generalized Kolmogorov criterion. Stoch. Process. Appl.
33, 29–44.

[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.
[26]
Seneta, E. (2006). Non-Negative Matrices and Markov Chains. Springer, New York.

[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.

[28]
Van Doorn, E. A. (1991). Quasi-stationary distributions and convergence to quasi-stationarity of birth-death processes. Adv. Appl. Prob.
23, 683–700.

[29]
Van Doorn, E. A. and Pollett, P. K. (2013). Quasi-stationary distributions for discrete-state models. Europ. J. Operat. Res.
230, 1–14.

[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.

[31]
Vere-Jones, D. (1967). Ergodic properties of nonnegative matrices. I. Pacific J. Math.
22, 361–386.

[32]
Villemonais, D. (2015). Minimal quasi-stationary distribution approximation for a birth and death process. Electron. J. Prob.
20, 30.

[33]
Woess, W. (2000). Random Walks on Infinite Graphs and Groups (Camb. Tracts Math. **138**). Cambridge University Press.

[34]
Woess, W. (2009). Denumerable Markov Chains: Generating Functions, Boundary Theory, Random Walks on Trees. European Mathematical Society, Zürich.