Discrete-time Markov chain: Difference between revisions

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Revision as of 12:26, 13 August 2020

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These descriptions highlight the structure of the Markov chain that is independent of the initial distribution <math>\Pr(X_1=x_1).</math> When time-homogeneous, the chain can be interpreted as a [[Finite-state machine|state machine]] assigning a probability of hopping from each vertex or state to an adjacent one. The probability <math>\Pr(X_n=x\mid X_1=x_1)</math> of the machine's state can be analyzed as the statistical behavior of the machine with an element <math>x_1</math> of the state space as input, or as the behavior of the machine with the initial distribution <math>\Pr(X_1=y)=[x_1=y]</math> of states as input, where <math>[P]</math> is the [[Iverson bracket]].{{cn|date=August 2020}}

==== Variations= ===

*{{Anchor|homogeneous}}Time-homogeneous Markov chains (or stationary Markov chains) are processes where

::<math>\Pr(X_{n+1}=x\mid X_n=y) = \Pr(X_n=x\mid X_{n-1}=y)</math>