Truncated hmcs Let P be a transition matrix on the countable
state space E, with the positive...

Question

Truncated hmcs Let P be a transition matrix on the countable
state space E, with the positive stationary distribution π. Let A be a
subset of the state space, and define the truncation of P on A to be the
transition matrix Q indexed by A and given by



Show that if (P, π) is reversible, then so is (Q, ).

Banasree · Accepted Answer

Ans. State space truncation 
Let Q(i,j) be a transition rate function on P that is reversible with respect to ᴨ.
If P is a CTMC on subset ᴨ ⸦ ᴨ (A)with transition rate Q(i.j) for  i ,j ꞓ P, then  (P, ᴨ ) is reversible with respect to Q restricted to ᴨ/ ᴨ(A).
In other word, a reversible process restricted to a subset of its state space is also reversible.

Truncated hmcs Let P be a transition matrix on the countable state space E, with the positive stationary distribution π. Let A be a subset of the state space, and define the truncation of P on A to be...

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