Markov chain, generalized

2020 Mathematics Subject Classification: Primary: 60J10 [MSN][ZBL]

A sequence of random variables with the properties:

1) the set of values of each is finite or countable;

2) for any and any ,

(*)

A generalized Markov chain satisfying (*) is called -generalized. For , (*) is the usual Markov property. The study of -generalized Markov chains can be reduced to the study of ordinary Markov chains. Consider the sequence of random variables whose values are in one-to-one correspondence with the values of the vector

The sequence forms an ordinary Markov chain.

References

[1]	J.L. Doob, "Stochastic processes" , Wiley (1953) MR1570654 MR0058896 Zbl 0053.26802

References

[a1]	D. Freedman, "Markov chains" , Holden-Day (1975) MR0686269 MR0681291 MR0556418 MR0428472 MR0292176 MR0237001 MR0211464 MR0164375 MR0158435 MR0152015 Zbl 0501.60071 Zbl 0501.60069 Zbl 0426.60064 Zbl 0325.60059 Zbl 0322.60057 Zbl 0212.49801 Zbl 0129.30605
[a2]	J.G. Kemeny, J.L. Snell, "Finite Markov chains" , v. Nostrand (1960) MR1531032 MR0115196 Zbl 0089.13704
[a3]	D. Revuz, "Markov chains" , North-Holland (1975) MR0415773 Zbl 0332.60045
[a4]	V.I. [V.I. Romanovskii] Romanovsky, "Discrete Markov chains" , Wolters-Noordhoff (1970) (Translated from Russian) MR0266312 Zbl 0201.20002
[a5]	E. Seneta, "Non-negative matrices and Markov chains" , Springer (1981) MR2209438 Zbl 0471.60001
[a6]	A. Blanc-Lapierre, R. Fortet, "Theory of random functions" , 1–2 , Gordon & Breach (1965–1968) (Translated from French) Zbl 0185.44502 Zbl 0159.45802