Some Inequalities for Reversible Markov Chains

Some Inequalities for Reversible Markov Chains

Generalized Cheeger inequalities for eigenvalues of non-reversible Markov chains

We show lower bounds for the smallest non-trivial eigenvalue, and smallest real portion of an eigenvalue, of the Laplacian of a non-reversible Markov chain in terms of an Evolving set quantity. A myriad of Cheeger-like inequalities follow for non-reversible chains, which even in the reversible case sharpen previously known results. The same argument also produces a new Cheeger-like inequality f...

Inequalities for Rare Events in Time-reversible Markov Chains I

The distribution of waiting time until a rare event is often approximated by the exponential distribution. In the context of rst hitting times for stationary reversible chains, the error has a simple explicit bound involving only the mean waiting time ET and the relaxation time of the chain. We recall general upper and lower bounds on ET and then discuss improvements available in the case ET wh...

Inequalities for Rare Events in Time-reversible Markov Chains Ii

A previous paper discussed explicit bounds in the exponential approximation for the distribution of the waiting time until a stationary reversible Markov chain rst enters a \rare" subset of states. In this paper Stein's method is used to get explicit (but complicated) bounds on the Poisson approximation for the number of non-adjacent visits to a rare subset.

Eigenvalues of non-reversible Markov chains: their connection to mixing times, reversible Markov chains, and Cheeger inequalities

We show a lower bound on mixing time for a non-reversible Markov chain in terms of its eigenvalues. This is used to show a bound on the real part of the complex-valued eigenvalues in terms of the realvalued eigenvalues of a related reversible chain, and likewise to bound the second largest magnitude eigenvalue. A myriad of Cheeger-like inequalities also follow for non-reversible chains, which e...