Related papers: Toy examples for effective concentration bounds
We derive the first explicit bounds for the spectral gap of a random walk Metropolis algorithm on $R^d$ for any value of the proposal variance, which when scaled appropriately recovers the correct $d^{-1}$ dependence on dimension for…
We consider evaluation of proper posterior distributions obtained from improper prior distributions. Our context is estimating a bounded function $\phi$ of a parameter when the loss is quadratic. If the posterior mean of $\phi$ is…
We consider periodic Markov chains with absorption. Applying to iterates of this periodic Markov chain criteria for the exponential convergence of conditional distributions of aperiodic absorbed Markov chains, we obtain exponential…
We prove that under mild positivity assumptions the entropy rate of a hidden Markov chain varies analytically as a function of the underlying Markov chain parameters. A general principle to determine the domain of analyticity is stated. An…
This paper derives exponential tail bounds and polynomial moment inequalities for the spectral norm deviation of a random matrix from its mean value. The argument depends on a matrix extension of Stein's method of exchangeable pairs for…
The partial sum of the states of a Markov chain or more generally a Markov source is asymptotically normally distributed under suitable conditions. One of these conditions is that the variance is unbounded. A simple combinatorial…
We describe a simple method that can be used to sample the rare fluctuations of discrete-time Markov chains. We focus on the case of Markov chains with well-defined steady-state measures, and derive expressions for the large-deviation rate…
We prove a functional limit theorem for Markov chains that, in each step, move up or down by a possibly state dependent constant with probability $1/2$, respectively. The theorem entails that the law of every one-dimensional regular…
This paper studies limit theorems for Markov Chains with general state space under conditions which imply subgeometric ergodicity. We obtain a central limit theorem and moderate deviation principles for additive not necessarily bounded…
Solutions to the Markov equation appear in many mathematical contexts. We aim to build on the understanding of them by proving a recent conjecture about Markov polynomials; solutions to a generalised version of the Markov equation. The…
We present a method for bounding, and in some cases computing, the spectral gap for systems of many particles evolving under the influence of a random collision mechanism. In particular, the method yields the exact spectral gap in a model…
In this paper, we develop necessary and sufficient conditions for the validity of a martingale approximation for the partial sums of a stationary process in terms of the maximum of consecutive errors. Such an approximation is useful for…
We consider the computational task of sampling a bit string $x$ from a distribution $\pi(x)=|\langle x|\psi\rangle|^2$, where $\psi$ is the unique ground state of a local Hamiltonian $H$. Our main result describes a direct link between the…
We prove explicit, i.e. non-asymptotic, error bounds for Markov chain Monte Carlo methods. The problem is to compute the expectation of a function f with respect to a measure {\pi}. Different convergence properties of Markov chains imply…
In this paper, we provide explicit lower bounds with respect to some quantities of interest (parameters of the underlying distribution, dimension, geometrical characteristics of the domain, position of the origin, etc.) on the spectral gap…
Matrix perturbation bounds play an essential role in the design and analysis of spectral algorithms. In this paper, we use a "contour bootstrapping" argument to derive several new perturbation bounds. As applications, we discuss new bounds…
We consider additive functionals of Markov processes in continuous time with general (metric) state spaces. We derive concentration bounds for their exponential moments and moments of finite order. Applications include diffusions,…
The main objective of this paper is to establish bootstrap uniform functional central limit theorem for Harris recurrent Markov chains over uniformly bounded classes of functions. We show that the result can be generalized also to the…
A discrete-time Markov chain can be transformed into a new Markov chain by looking at its states along iterations of an almost surely finite stopping time. By the optional stopping theorem, any bounded harmonic function with respect to the…
On a weakly Blackwell space we show how to define a Markov chain approximating problem, for the target problem. The approximating problem is proved to converge to the optimal reduced problem under different pseudometrics. A computational…