Markov kernel: Difference between revisions

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Revision as of 14:51, 30 November 2022

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== Examples ==

===[[Simple random walk]] on the integers ===

Take <math>X=Y=\Z</math>, and <math> \mathcal A = \mathcal B = \mathcal P(\Z)</math> (the [[power set]] of <math>\Z</math>). Then a Markov kernel is fully determined by the probability it assigns to a singleton setsingletons <math>\{m\}</math>,\, with <math>m \in Y = \Z</math> for each <math>n \in X = \Z</math>:

:<math>\kappa(B|n )=\sum_{m \in B}\kappa(\{m\}|n), \qquad \forall n \in \mathbb{Z}, \, \forall B \in \mathcal B</math>.

Now the random walk <math>\kappa</math> that goes to the right with probability <math>p</math> and to the left with probability <math>1 - p</math> is defined by