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Generalized gamma distributions arise as limits in many settings involving random graphs, walks, trees, and branching processes. Pek\"oz, R\"ollin, and Ross (2016, arXiv:1309.4183 [math.PR]) exploited characterizing distributional fixed…
For many stochastic diffusion processes with mean field interaction, convergence of the rescaled total mass processes towards a diffusion process is known. Here we show convergence of the so-called finite system scheme for interacting…
We study an approximation by time-discretized geodesic random walks of a diffusion process associated with a family of time-dependent metrics on manifolds. The condition we assume on the metrics is a natural time-inhomogeneous extension of…
This paper presents a self-contained account for coupling arguments and applications in the context of Markov processes. We first use coupling to describe the transport problem, which leads to the concepts of optimal coupling and…
We establish a novel convergent iteration framework for a weak approximation of general switching diffusion. The key theoretical basis of the proposed approach is a restriction of the maximum number of switching so as to untangle and…
By developing a new technique called the bi-coupling argument, we estimate the relative entropy between different diffusion processes in terms of the distances of initial distributions and drift-diffusion coefficients. As an application,…
We prove Berry-Esseen type rates of convergence for central limit theorems (CLTs) of regenerative processes which generalize previous results of Bolthausen under weaker moment assumptions. We then show how this general result can be applied…
We construct a renewal structure for random walks on surface groups. The renewal times are defined as times when the random walks enters a particular type of a cone and never leaves it again. As a consequence, the trajectory of the random…
We obtain a Lundberg-type inequality in the case of an inhomogeneous renewal risk model. We consider the model with independent, but not necessarily identically distributed, claim sizes and the interoccurrence times. In order to prove the…
Enriching Brownian motion with regenerations from a fixed regeneration distribution $\mu$ at a particular regeneration rate $\kappa$ results in a Markov process that has a target distribution $\pi$ as its invariant distribution. For the…
In this article we extend the coupling method from classical probability theory to quantum Markov chains on atomic von Neumann algebras. In particular, we establish a coupling inequality, which allow us to estimate convergence rates by…
We present a novel approach to quantizing Markov chains. The approach is based on the Markov chain coupling method, which is frequently used to prove fast mixing. Given a particular coupling, e.g., a grand coupling, we construct a…
Our purpose is to obtain gradient estimates for certain nonlinear partial differential equations by coupling methods. First we derive uniform gradient estimates for a certain semi-linear PDEs based on the coupling method introduced in Wang…
The new class of alternating-conjugate splitting methods is presented and analyzed. They are obtained by concatenating a given composition involving complex coefficients with the same composition but with the complex conjugate coefficients.…
We propose a coupled rejection-sampling method for sampling from couplings of arbitrary distributions. The method relies on accepting or rejecting coupled samples coming from dominating marginals. Contrary to existing acceptance-rejection…
We describe a novel switching algorithm based on a ``reverse'' Monte Carlo method, in which the potential is stochastically modified before the system configuration is moved. This new algorithm facilitates a generalized formulation of…
The purpose of this dissertation is to introduce a version of Stein's method of exchangeable pairs to solve problems in measure concentration. We specifically target systems of dependent random variables, since that is where the power of…
Convergence problems in coupled-cluster iterations are discussed, and a new iteration scheme is proposed. Whereas the Jacobi method inverts only the diagonal part of the large matrix of equation coefficients, we invert a matrix which also…
We present a new and simple approach to concentration inequalities for functions around their expectation with respect to non-product measures, i.e., for dependent random variables. Our method is based on coupling ideas and does not use…
This paper is concerned with elucidating a relationship between two common coupling methods for the continuous time Markov chain models utilized in the cell biology literature. The couplings considered here are primarily used in a…