Is the variance of an eigenfunction of a finite state space aperiodic irreducible markov chain starting at a single state always non-decreasing? - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-18T08:02:47Z http://mathoverflow.net/feeds/question/58275 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/58275/is-the-variance-of-an-eigenfunction-of-a-finite-state-space-aperiodic-irreducible Is the variance of an eigenfunction of a finite state space aperiodic irreducible markov chain starting at a single state always non-decreasing? John Jiang 2011-03-12T18:12:39Z 2011-03-13T00:07:49Z <p>I am reposting a previous question due to incorrect initial formulation. </p> <p>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.</p> <p>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. </p>