Markov chain, generalized

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

A sequence of random variables $ \xi _ {n} $ with the properties:

1) the set of values of each $ \xi _ {n} $ is finite or countable;

2) for any $ n $ and any $ i _ {0} \dots i _ {n} $,

$$ \tag{* } {\mathsf P} \{ \xi _ {n} = i _ {n} \mid \xi _ {0} = i _ {0} \dots \xi _ {n-} s = i _ {n-} s \dots \xi _ {n-} 1 = i _ {n-} 1 \} = $$

$$ = \ {\mathsf P} \{ \xi _ {n} = i _ {n} \mid \xi _ {n-} s = i _ {n-} s \dots \xi _ {n-} 1 = i _ {n-} 1 \} . $$

A generalized Markov chain satisfying (*) is called $ s $- generalized. For $ s = 1 $, (*) is the usual Markov property. The study of $ s $- generalized Markov chains can be reduced to the study of ordinary Markov chains. Consider the sequence of random variables $ \eta _ {n} $ whose values are in one-to-one correspondence with the values of the vector

$$ ( \xi _ {n-} s+ 1 , \xi _ {n-} s+ 2 \dots \xi _ {n} ) . $$

The sequence $ \eta _ {n} $ forms an ordinary Markov chain.

References

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

References

[F]	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
[KS]	J.G. Kemeny, J.L. Snell, "Finite Markov chains" , v. Nostrand (1960) MR1531032 MR0115196 Zbl 0089.13704
[Re]	D. Revuz, "Markov chains" , North-Holland (1975) MR0415773 Zbl 0332.60045
[Ro]	V.I. Romanovsky, "Discrete Markov chains" , Wolters-Noordhoff (1970) (Translated from Russian) MR0266312 Zbl 0201.20002
[S]	E. Seneta, "Non-negative matrices and Markov chains" , Springer (1981) MR2209438 Zbl 0471.60001
[BF]	A. Blanc-Lapierre, R. Fortet, "Theory of random functions" , 1–2 , Gordon & Breach (1965–1968) (Translated from French) Zbl 0185.44502 Zbl 0159.45802