What is the maximum number of possible paths in an absorbing markov chain starting form a random transient state?
2026-03-24 23:45:54.1774395954
Maximum number of possible paths in an absorbing markov chain
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In the worst case, consider your Markov chain as a triangle, with vertices $A,B,C$ and $A$ absorbing, with all other transitions of equal weight (i.e. moving from $B$ or from $C$, you end up at $A$ with probability $1/2$).
It is clear that even in this simple construction, there is a (countably) infinite number of possible paths, e.g. BA, BCA, BCBA, BCBCA, etc.
Note that each path of bigger length ends up exponentially less probable, which is what makes the final distribution collapse to $(p_A, p_B, p_C) = (1,0,0)$.
Of course, starting at $C$, the situation will be symmetric...