Deviation inequalities for bifurcating Markov chains on Galton−Watson tree

Large Deviation Bounds for Markov Chains

We study the fraction of time that a Markov chain spends in a given subset of states. We give an exponential bound on the probability that it exceeds its expectation by a constant factor. Our bound depends on the mixing properties of the chain, and is asymptotically optimal for a certain class of Markov chains. It beats the best previously known results in this direction. We present an applicat...

Cheeger inequalities for transient Markov chains

We construct upper and lower bounds for Cheeger-type constants for transient Markov chains. We first treat the situation where there is a detailed balance condition and obtain bounds that rely exclusively on the first and second eigenvalues of the substochastic transition matrix. Secondly, we consider general substochastic transition matrices and develop bounds in this broader setting. The effi...

Some Inequalities for Reversible Markov Chains

Nash Inequalities for Finite Markov Chains

This paper develops bounds on the rate of decay of powers of Markov kernels on finite state spaces. These are combined with eigenvalue estimates to give good bounds on the rate of convergence to stationarity for finite Markov chains whose underlying graph has moderate volume growth. Roughly, for such chains, order (diameter) 2 steps are necessary and suffice to reach stationarity. We consider l...

Comparison Inequalities and Fastest-mixing Markov Chains

We introduce a new partial order on the class of stochastically monotone Markov kernels having a given stationary distribution π on a given finite partially ordered state space X . When K L in this partial order we say that K and L satisfy a comparison inequality. We establish that if K1, . . . ,Kt and L1, . . . , Lt are reversible and Ks Ls for s = 1, . . . , t, then K1 · · ·Kt L1 · · ·Lt. In ...