Markov Chain - Time Reversibility proof

[Jimmy Zhan]

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 x_iy_iπ_i, 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

q_ij = p_ij
where q_ij is the transition probability from state i to state j in the reverse chain.
p_ij = p_jiπ_j / π_i
π_ip_ij = π_jp_ji.

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.

 

[Ray Vickson]

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 x_iy_iπ_i, 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

q_ij = p_ij
where q_ij is the transition probability from state i to state j in the reverse chain.
p_ij = p_jiπ_j / π_i
π_ip_ij = π_jp_ji.

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.

You cannot receive help until you show your work on the problem. What have you actually done? Don't just describe it in words; write down formulas, equations, etc. Perhaps you wrote incorrect formulas; we cannot know if you don't show us.

 

[Jimmy Zhan]

The attempt at solution:

⟨x, Py⟩ = ⟨Px, y⟩
∑_ix_i(Py)π_i = ∑_i(Px)_iy_iπ_i
∑_i x_i [Py]_i π_i = ∑_i [Px]_i y_i π_i

However I don't know what is the form of the P matrix and thus the form of the matrix product Px and Py.

 

[Ray Vickson]

The attempt at solution:

⟨x, Py⟩ = ⟨Px, y⟩
∑_ix_i(Py)π_i = ∑_i(Px)_iy_iπ_i
∑_i x_i [Py]_i π_i = ∑_i [Px]_i y_i π_i

However I don't know what is the form of the P matrix and thus the form of the matrix product Px and Py.

You don't know the formula for the components of $\mathbf{Px}$? Then how can you have studied anything about Markov chains, where those formulas are used extensively?