2,241 research outputs found
A Linear Programming Approach to Error Bounds for Random Walks in the Quarter-plane
We consider the approximation of the performance of random walks in the
quarter-plane. The approximation is in terms of a random walk with a
product-form stationary distribution, which is obtained by perturbing the
transition probabilities along the boundaries of the state space. A Markov
reward approach is used to bound the approximation error. The main contribution
of the work is the formulation of a linear program that provides the
approximation error
An approximation scheme for quasi-stationary distributions of killed diffusions
In this paper we study the asymptotic behavior of the normalized weighted
empirical occupation measures of a diffusion process on a compact manifold
which is killed at a smooth rate and then regenerated at a random location,
distributed according to the weighted empirical occupation measure. We show
that the weighted occupation measures almost surely comprise an asymptotic
pseudo-trajectory for a certain deterministic measure-valued semiflow, after
suitably rescaling the time, and that with probability one they converge to the
quasi-stationary distribution of the killed diffusion. These results provide
theoretical justification for a scalable quasi-stationary Monte Carlo method
for sampling from Bayesian posterior distributions.Comment: v2: revised version, 29 pages, 1 figur
Quantum logarithmic Sobolev inequalities and rapid mixing
A family of logarithmic Sobolev inequalities on finite dimensional quantum
state spaces is introduced. The framework of non-commutative \bL_p-spaces is
reviewed and the relationship between quantum logarithmic Sobolev inequalities
and the hypercontractivity of quantum semigroups is discussed. This
relationship is central for the derivation of lower bounds for the logarithmic
Sobolev (LS) constants. Essential results for the family of inequalities are
proved, and we show an upper bound to the generalized LS constant in terms of
the spectral gap of the generator of the semigroup. These inequalities provide
a framework for the derivation of improved bounds on the convergence time of
quantum dynamical semigroups, when the LS constant and the spectral gap are of
the same order. Convergence bounds on finite dimensional state spaces are
particularly relevant for the field of quantum information theory. We provide a
number of examples, where improved bounds on the mixing time of several
semigroups are obtained; including the depolarizing semigroup and quantum
expanders.Comment: Updated manuscript, 30 pages, no figure
Gibbs point process approximation: Total variation bounds using Stein's method
We obtain upper bounds for the total variation distance between the
distributions of two Gibbs point processes in a very general setting.
Applications are provided to various well-known processes and settings from
spatial statistics and statistical physics, including the comparison of two
Lennard-Jones processes, hard core approximation of an area interaction process
and the approximation of lattice processes by a continuous Gibbs process. Our
proof of the main results is based on Stein's method. We construct an explicit
coupling between two spatial birth-death processes to obtain Stein factors, and
employ the Georgii-Nguyen-Zessin equation for the total bound.Comment: Published in at http://dx.doi.org/10.1214/13-AOP895 the Annals of
Probability (http://www.imstat.org/aop/) by the Institute of Mathematical
Statistics (http://www.imstat.org
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