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→Simple random walk on the integers: Add forall B \in \mathcal B in singleton to arbitrary |
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===[[Simple random walk]] on the integers ===
Take <math>X=Y=\Z, \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 singleton sets 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
:<math>\kappa(\{m\}|n)= p \delta_{m, n + 1}+ (1-p) \delta_{m, n - 1}, \quad \forall n,m \in \Z</math>
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