Related papers: Quantitative convergence rates for reversible Mark…

Dealing with finite Markov chains in discrete time, the focus often lies on convergence behavior and one tries to make different copies of the chain meet as fast as possible and then stick together. There is, however, a very peculiar kind…

We establish the weak large deviations principle for empirical measures of Markov chains on $\mathbb R^d$ under mild assumptions. In particular, no irreducibility is assumed and the initial measure may be arbitrary. The proof is entirely…

We study finite-time spectral rigidity in reversible Markov chains via exact spectral relaxation dynamics. While the underlying identities follow classically from self-adjointness on $L^2(\pi)$, organizing the dynamics around the relaxation…

The switch chain is a well-known Markov chain for sampling directed graphs with a given degree sequence. While not ergodic in general, we show that it is ergodic for regular degree sequences. We then prove that the switch chain is rapidly…

This paper considers how to obtain MCMC quantitative convergence bounds which can be translated into tight complexity bounds in high-dimensional {settings}. We propose a modified drift-and-minorization approach, which establishes…

We produce the first example of bounding total variation distance to stationarity and estimating mixing times via orthogonal polynomials diagonalization of discrete reversible Markov chains, the Karlin-McGregor approach.

We develop Markov chain mixing time estimates for a class of Markov chains with restricted transitions. We assume transitions may occur along a cycle of $n$ nodes and on $n^\gamma$ additional edges, where $\gamma < 1$. We find that the…

Suppose X and Y are two independent irreducible Markov chains on n states. We consider the intersection time, which is the first time their trajectories intersect. We show for reversible and lazy chains that the total variation mixing time…

We consider triangular arrays of Markov chains that converge weakly to a diffusion process. Local limit theorems for transition densities are proved. The observation time [0,T] may be fixed or lim n T = 0, where nh = T and h is a mesh…

Recursive stochastic algorithms have gained significant attention in the recent past due to data driven applications. Examples include stochastic gradient descent for solving large-scale optimization problems and empirical dynamic…

We consider continuous-time Markov chains on integers which allow transitions to adjacent states only, with alternating rates. We give explicit formulas for probability generating functions, and also for means, variances and state…

We extend our previous study of Markov chains on finite commutative rings (arXiv:1605.05089) to arbitrary finite rings with identity. At each step, we either add or multiply by a randomly chosen element of the ring, where the addition…

We introduce quantum versions of the $\chi^2$-divergence, provide a detailed analysis of their properties, and apply them in the investigation of mixing times of quantum Markov processes. An approach similar to the one presented in [1-3]…

We prove several results about the rate of convergence to stationarity, that is, the spectral gap, for the M/M/n queue in the Halfin-Whitt regime. We identify the limiting rate of convergence to steady-state, and discover an asymptotic…

We study the limit behaviour of a generally non-linear ordinary differential equation whose solution is a superadditive generalisation of a stochastic matrix, and provide necessary and sufficient conditions for this solution to be ergodic,…

In this paper we extend the results of the research started by the first author, in which Karlin-McGregor diagonalization of certain reversible Markov chains over countably infinite general state spaces by orthogonal polynomials was used to…

We propose a time-independent Hamiltonian protocol for the reversal of qubit ordering in a chain of $N$ spins. Our protocol has an easily implementable nearest-neighbor, transverse-field Ising model Hamiltonian with time-independent,…

We analyse the $\ell^2(\pi)$-convergence rate of irreducible and aperiodic Markov chains with $N$-band transition probability matrix $P$ and with invariant distribution $\pi$. This analysis is heavily based on: first the study of the…

For a reversible and ergodic Markov chain $\{X_n,n\geq0\}$ with invariant distribution $\pi$, we show that a valid confidence interval for $\pi(h)$ can be constructed whenever the asymptotic variance $\sigma^2_P(h)$ is finite and positive.…

We present a recurrence-transience classification for discrete-time Markov chains on manifolds with negative curvature. Our classification depends only on geometric quantities associated to the increments of the chain, defined via the…