On the rate of convergence of the Gibbs sampler for the 1-D Ising model by geometric bound

In this paper, we are concerned with the simulation of Gaussian random fields by means of iterative stochastic algorithms, which are compared in terms of rate of convergence. A parametrized class of algorithms, which includes stochastic relaxation (Gibbs sampler), is proposed and its convergence properties are established. A suitable choice for the parameter improves the rate of convergence with respect to stochastic relaxation for special classes of covariance matrices. Some examples and numerical experiments are given.

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On the method of Yvon in crystal statistics

Proceedings of the Royal Society of London Series A - Mathematical and Physical Sciences ◽

10.1098/rspa.1962.0155 ◽

1962 ◽

Vol 268 (1335) ◽

pp. 506-526 ◽

Cited By ~ 29

Keyword(s):

Ising Model ◽

Closed Form ◽

Rate Of Convergence ◽

Series Expansions ◽

Partition Functions ◽

Integral Type ◽

Insufficient Number

Closed-form approximations in crystal statistics have suffered from the defect that no steady series of approximations was available from which the rate of convergence could be assessed. The method of Yvon, based on a cluster-integral type of development, can furnish such a series of approximations. It is shown how to construct the partition functions for such approximations, the key being the use of the Mayer theory for multi-component assemblies. The method is applied to various properties of the Ising model of a ferromagnet and antiferromagnet, and results are obtained which are consistent with those based on series expansions. Previous investigations of Fournet which gave different results are shown to have made use of an insufficient number of terms in the approximation.

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A Bound on the Rate of Convergence for the Discrete Gibbs Sampler

Probability in the Engineering and Informational Sciences ◽

10.1017/s0269964800003806 ◽

1995 ◽

Vol 9 (2) ◽

pp. 211-215 ◽

Cited By ~ 1

Author(s):

I. H. Dinwoodie

Keyword(s):

Rate Of Convergence ◽

Stationary Distribution ◽

Gibbs Sampler ◽

Occupation Measure ◽

Computable Bound

We give a computable bound on the rate of convergence of the occupation measure for the Gibbs sampler to the stationary distribution.

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Products of 2 × 2 stochastic matrices with random entries

Journal of Applied Probability ◽

10.1017/s0021900200115943 ◽

1986 ◽

Vol 23 (04) ◽

pp. 1019-1024

Author(s):

Walter Van Assche

Keyword(s):

Rate Of Convergence ◽

Beta Distribution ◽

Stopping Time ◽

Stochastic Matrices

The limit of a product of independent 2 × 2 stochastic matrices is given when the entries of the first column are independent and have the same symmetric beta distribution. The rate of convergence is considered by introducing a stopping time for which asymptotics are given.

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