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Determine Markov chain asymptotics

[1]
Gallager, R.G. *Stochastic Processes: Theory for Applications.* Cambridge, UK: Cambridge University Press, 2013.

[2]
Horn, R., and C. R.
Johnson. *Matrix Analysis.* Cambridge, UK: Cambridge University Press,
1985.

[3]
Seneta, E. *Non-negative Matrices and Markov Chains.* New York, NY: Springer-Verlag, 1981.