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Markov Shift

A Markov shift is the shift on the sequence space of a stationary Markov chain, equipped with its stationary probability measure. Let S be a countable set of states, let P be a stochastic matrix on S, and let pi be a stationary statistical distribution, so that piP=pi. On S^({0,1,2,...}), the Markov probability measure mu is specified by

mu{x:x_0=i_0,...,x_n=i_n}=pi_(i_0)P_(i_0i_1)...P_(i_(n-1)i_n).

The sigma-algebra is generated by these finite-coordinate sets. The shift is the map

sigma(x_0,x_1,x_2,...)=(x_1,x_2,x_3,...),

and mu is an invariant measure for sigma (Walkden 2018, pp. 34-35).

See also

Markov Chain

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References

Walkden, C. Ergodic Theory. Lecture notes, Jan. 4, 2018. https://personalpages.manchester.ac.uk/staff/Charles.Walkden/ergodic-theory/ergodic_theory.pdf.

Cite this as:

Weisstein, Eric W. "Markov Shift." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/MarkovShift.html

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