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A question from Brémaud, Markov Chains Gibbs Fields, Monte Carlo Simulation and Queues.

Exercise 11.5.2. Metropolis and Barker Algorithms: Irreducibility

how do you show explicitly that $$p_{i,j}= \alpha (i,j)q(i,j)$$ $$p_{i,i}=1-\sum_{k:k\neq i}\alpha (i,k)q(i,k)$$

is irreducible where enter image description here

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