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I am reposting a previous question due to incorrect initial formulation.

Given an ergodic (aperiodic and irreducible) finite state space Markov chain $P$. Let $f$ be an eigenfunction, i.e., $P_t f = \lambda^t f$. Let the chain start at a single state $x \in \Omega$. Is it true that $\mathbb{E}_x (P_tf- \mathbb{E}_x( P_t f))^2$ is nondecreasing in t? As pointed out by James Martin, it's important to start with the point mass distribution at a single state, since otherwise one could choose an initial distribution that has an f-variance larger than the stationary $f$-variance.

Wilson's method gives a way to bound the variance of eigenfunctions in t in a way that's reminiscent of the Martingale difference method, but since the variance at time \infty $\infty$ is usually easy to calculate, if we know the above monotonicity result, we could bound the variance at finite time easily.

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I am reposting a previous question due to incorrect initial formulation.

Given an ergodic (aperiodic and irreducible) finite state space Markov chain $P$. Let $f$ be an eigenfunction, i.e., $P_t f = \lambda^t f$. Let the chain start at a single state $x \in \Omega$. Is it true that \mathbb{E}_x $\mathbb{E}_x (P_tf- \mathbb{E}_x( P_t f))^2 f))^2$ is nondecreasing in t? As pointed out by James Martin, it's important to start with the point mass distribution at a single state, since otherwise one could choose an initial distribution that has an f-variance larger than the stationary $f$-variance.

Wilson's method gives a way to bound the variance of eigenfunctions in t in a way that's reminiscent of the Martingale difference method, but since the variance at time \infty is usually easy to calculate, if we know the above monotonicity result, we could bound the variance at finite time easily.

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# Is the variance of an eigenfunction of a finite state space aperiodic irreducible markov chain starting at a single state always non-decreasing?

I am reposting a previous question due to incorrect initial formulation.

Given an ergodic (aperiodic and irreducible) finite state space Markov chain $P$. Let $f$ be an eigenfunction, i.e., $P_t f = \lambda^t f$. Let the chain start at a single state $x \in \Omega$. Is it true that \mathbb{E}_x (P_tf- \mathbb{E}_x( P_t f))^2 is nondecreasing in t? As pointed out by James Martin, it's important to start with the point mass distribution at a single state, since otherwise one could choose an initial distribution that has an f-variance larger than the stationary $f$-variance.

Wilson's method gives a way to bound the variance of eigenfunctions in t in a way that's reminiscent of the Martingale difference method, but since the variance at time \infty is usually easy to calculate, if we know the above monotonicity result, we could bound the variance at finite time easily.