SIMILAR MARKOV CHAINS
by
Phil Pollett
Department of Mathematics
The University of Queensland
There is an accompanying paper:
Pollett, P.K. (2001) Similar Markov chains. Journal of Applied Probability 38A (to appear). 
From that paper:
"I was introduced to David VereJones' work by another of my mentors, Charles Pearce, whose lively undergraduate lectures on the VereJones theory did much to convince me that I should follow a probability path. David's work has been a constant source of inspiration to me and I continue to benefit from our fruitful and enjoyable collaborations. I am grateful to him personally for his advice, encouragement and friendship over many years and I therefore take great pleasure in dedicating this short note to him." 
From those lectures:
MAIN REFERENCES
Convergence of Markov transition probabilities and their spectral properties
1. VereJones, D. Geometric ergodicity in denumerable Markov chains. Quart. J. Math. Oxford Ser. 2 13 (1962) 728.
2. VereJones, D. On the spectra of some linear operators associated with queueing systems. Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 2 (1963) 1221.
3. VereJones, D. Ergodic properties of nonnegative matrices. I. Pacific J. Math. 22 (1967) 361386.
4. VereJones, D. Ergodic properties of nonnegative matrices. II. Pacific J. Math. 26 (1968) 601620.
Classification of transient Markov chains and quasistationary distributions
5. Seneta, E.; VereJones, D. On quasistationary distributions in discretetime Markov chains with a denumerable infinity of states. J. Appl. Probability 3 (1966) 403434.
6. VereJones, D. Some limit theorems for evanescent processes. Austral. J. Statist. 11 (1969) 6778.
Related work
7. VereJones, D.; Kendall, David G. A commutativity problem in the theory of Markov chains. Teor. Veroyatnost. i Primenen. 4 (1959) 97100.
8. VereJones, D. A rate of convergence problem in the theory of queues. Teor. Verojatnost. i Primenen. 9 (1964) 104112.
9. VereJones, D. Note on a theorem of Kingman and a theorem of Chung. Ann. Math. Statist. 37 (1966) 18441846.
10. Heathcote, C. R.; Seneta, E.; VereJones, D. A refinement of two theorems in the theory of branching processes. Teor. Verojatnost. i Primenen. 12 (1967) 341346.
11. Rubin, H.; VereJones, D. Domains of attraction for the subcritical GaltonWatson branching process. J. Appl. Probability 5 (1968) 216219.
12. Seneta, E.; VereJones, D. On the asymptotic behaviour of subcritical branching processes with continuous state space. Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 10 (1968) 212225.
13. Fahady, K. S.; Quine, M. P.; VereJones, D. Heavy traffic approximations for the GaltonWatson process. Advances in Appl. Probability 3 (1971) 282300.
14. Pollett, P. K.; VereJones, D. A note on evanescent processes. Austral. J. Statist. 34 (1992), no. 3, 531536.
Important early work on quasistationary distributions
Yaglom, A.M. Certain limit theorems of the theory of branching processes (Russian) Doklady Akad. Nauk SSSR (N.S.) 56 (1947) 795798.
Bartlett, M.S. Deterministic and stochastic models for recurrent epidemics. Proceedings of the Third Berkeley Symposium on Mathematical Statistics and Probability, 19541955, Vol. IV, pp. 81109. University of California Press, Berkeley and Los Angeles, 1956.
Bartlett, M.S. Stochastic population models in ecology and epidemiology. Methuen's Monographs on Applied Probability and Statistics Methuen & Co., Ltd., London; John Wiley & Sons, Inc., New York, 1960.
Darroch, J. N.; Seneta, E. On quasistationary distributions in absorbing discretetime finite Markov chains. J. Appl. Probability 2 (1965) 88100.
Darroch, J. N.; Seneta, E. On quasistationary distributions in absorbing continuoustime finite Markov chains. J. Appl. Probability 4 (1967) 192196.
Important early work on quasistationary distributions
Mandl, Petr Sur le comportement asymptotique des probabilités dans les ensembles des états d'une chaîne de Markov homogène (Russian) Casopis Pest. Mat. 84 (1959) 140149.
Mandl, Petr On the asymptotic behaviour of probabilities within groups of states of a homogeneous Markov process (Czech) Casopis Pest. Mat. 85 (1960) 448456.
Ewens, W.J. The diffusion equation and a pseudodistribution in genetics. J. Roy. Statist. Soc., Ser B 25 (1963) 405412.
Kingman, J.F.C. The exponential decay of Markov transition probabilities. Proc. London Math. Soc. 13 (1963) 337358.
Ewens, W.J. The pseudotransient distribution and its uses in genetics. J. Appl. Probab. 1 (1964) 141156.
Seneta, E. Quasistationary distributions and timereversion in genetics. (With discussion) J. Roy. Statist. Soc. Ser. B 28 (1966) 253277.
Seneta, E. Quasistationary behaviour in the random walk with continuous time. Austral. J. Statist. 8 (1966) 9298.
DISCRETETIME CHAINS
Setting: , a timehomogeneous Markov chain taking values in a countable set S with transition probabilities
Let C be any irreducible and (for simplicity) aperiodic class.
DVJ1: For , as . The limit does not depend on i and it satisfies . Moreover, and indeed, for , , where .
(If C is recurrent, implies . When C is transient, we can have , or, , which is called geometric ergodicity.)
DVJ2: For any real r>0, the series , , converge or diverge together; in particular, they have the same radius of convergence R, and . And, all or none of the sequences tend to zero.
Figure 1. Simulation of a metapopulation model.
(Download the following image (as postscript)):
Figure 2. State transition diagram for
epidemic model.
(Download the following image (as postscript)):
Figure 3. Simulation of the epidemic model
[Simulation 
java code]
(Download the following image (as postscript)):
Figure 4. Autocatalytic reaction simulation
[Simulation 
java code]
(Download the following image (as postscript)):
Figure 5. Quasistationarity of an explosive population process
[Simulation

java code]
TRANSIENT CHAINS
The key to unlocking this "quasistationarity" is to examine the behaviour of the transition probabilities at the radius of convergence R.
Suppose that C is transient class which is geometrically ergodic ( , R>1). Although , it might be true that , where . How does this help?
For ,
provided that we can justify taking limit under summation.
DVJ3: C is said to be Rtransient or Rrecurrent according as converges or diverges. If C is Rrecurrent, then it is said to be Rpositive or Rnull according to whether the limit of is positive or zero.
DVJ4: If C is Rrecurrent, then, for , the inequalities
have unique positive solutions and and indeed they are eigenvectors:
C is then Rpositive recurrent if and only if , in which case
and, if , then
AND FINALLY
SDVJ: If C is Rpositive recurrent and the lefteigenvector satisfies , then the limiting conditional (or quasistationary) distribution exists: as ,
... AND MUCH MORE
Other kinds of QSD, more general and more precise statements, continuoustime chains, general state spaces, numerical methods and in particular truncation methods, MCMC, countless applications of QSDs: chemical kinetics, population biology, ecology, epidemiology, reliability, telecommunications. A full bibliography is maintained at my web site:
http://www.maths.uq.edu.au/~pkp/research.html
(Download the following image (as postscript)):
Figure 6. Surface plot of the quasistationary distribution
for the epidemic model
SIMILAR MARKOV CHAINS
New setting: , a timehomogeneous Markov chain in continuous time taking values in a countable set S, with transition function , where
Assuming that (standard), the transitions rates are defined by . Set and assume (stable).
Definition: Two such chains X and are said to be similar if their transition functions, P and , satisfy , , t>0, for some collection of positive constants , .
Immediate consequences of the definition: Since both chains are standard, and the transition rates must satisfy , in particular, . They share the same irreducible classes and the same classification of states.
Birthdeath chains: Lenin et al. proved that for birthdeath chains the "similarity constants" must factorize as . (Note that , since .)
Is this true more generally?
Definition: Let C be a subset of S. Two chains are said to be strongly similar over C if , , t>0, for some collection of positive constants , .
Proposition: If C is recurrent, then .
(Proof: .)
EXTENSION OF DJV THEORY
Kingman: If C is irreducible, then, for each , ( ), , for some , et cetera.
Definition: C is said to be transient or recurrent according as converges or diverges. If C is recurrent, then it is said to be positive or null according to whether the limit of is positive or zero.
Theorem: If C is recurrent, then, for , the inequalities
have unique positive solutions and and indeed they are eigenvectors:
C is then positive recurrent if and only if , in which case
Suppose that P and are similar. They share the same and the same "classification".
Theorem: If C is a positive recurrent class, then P and are strongly similar over C. We may take , where and are the essentially unique invariant measures (left eigenvectors) on C for P and for , respectively.
Proof: Let in . We get, in an obvious notation,
and, since , we have .
Again: Are similar chains always strongly similar?
In the null recurrent case, it may still be possible to deduce the desired factorization, for, although , it may be possible to find a such that tends to a strictly positive limit. (Similar chains will have the same .)
Lemma: Assume that C is null recurrent and suppose that there is a , which does not depend on i and j, such that tends to a strictly positive limit for all . Then, there is a positive constant d such that , , where and are, respectively, the essentially unique invariant measure and vector (left and righteigenvectors) on C for P.
Remark: Even in the transient case it might still be possible to find a such that tends to a positive limit, and for the conclusions to the lemma to hold good. (Note that, by the usual irreducibility arguments, will be the same for all i and j in any given class.)