- #1
Jimmy Zhan
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Homework Statement
Let X = {Xn : n ≥ 0} be an irreducible, aperiodic Markov chain with finite state space S, transition matrix P, and stationary distribution π. For x,y ∈ R|S|, define the inner product ⟨x,y⟩ = ∑i∈S xiyiπi, and let L2(π) = {x ∈ R|S| : ⟨x,x⟩ < ∞}. Show that X is time-reversible if and only if ⟨x, Py⟩ = ⟨Px, y⟩ for all x, y ∈ L2(π).
Homework Equations
X is reversible if and only if
qij = pij
where qij is the transition probability from state i to state j in the reverse chain.
pij = pjiπj / πi
πipij = πjpji.
The Attempt at a Solution
I tried equating ⟨x, Py⟩ = ⟨Px, y⟩ and subbing in the definition of the inner product as defined in the question. However, that doesn't seem to lead to anywhere.
Thank you for all those who can help.