Abstract. Iterated random functions are used to draw pictures or simulate large Ising models, among other applications. They offer a method for studying the steady state distribution of a Markov chain, and give useful bounds on rates of convergence in a variety of examples. The present paper surveys the field and presents some new examples. There is a simple unifying idea: the iterates of random Lipschitz functions converge if the functions are contracting on the average. 1. Introduction. The

For a large class of examples arising in statistical physics known as attractive spin systems (e.g., the Ising model), one seeks to sample from a probability distribution # on an enormously large state space, but elementary sampling is ruled out by the infeasibility of calculating an appropriate normalizing constant. The same difficulty arises in computer science problems where one seeks to sample randomly from a large finite distributive lattice whose precise size cannot be ascertained in any reasonable amount of time. The Markov chain Monte Carlo (MCMC) approximate sampling approach to such a problem is to construct and run "for a long time" a Markov chain with long-run distribution #. But determining how long is long enough to get a good approximation can be both analytically and empirically difficult. Recently, Jim Propp and David Wilson have devised an ingenious and efficient algorithm to use the same Markov chains to produce perfect (i.e., exact) samples from #. However, the running t...

A new approach to Markov chain Monte Carlo simulation was recently proposed by Propp and Wilson. This approach, unlike traditional ones, yields samples which have exactly the desired distribution. The Propp-Wilson algorithm requires this distribution to have a certain structure called monotonicity. In this paper an idea of Kendall is applied to show how the algorithm can be extended to the case where monotonicity is replaced by anti-monotonicity. As illustrating examples, simulations of the hard-core model and the random-cluster model are presented.

We present a perfect simulation algorithm for measures that are absolutely continuous with respect to some Poisson process and can be obtained as invariant measures of birth-and-death processes. Examples include area- and perimeter-interacting point processes (with stochastic grains), invariant measures of loss networks, and the Ising contour and random cluster models. The algorithm does not involve any coupling ---hence it is not tied up to monotonicity requirements--- and it directly provides samples of the infinite-volume measure. The algorithm is based on a two-step procedure: (i) a perfect-simulation scheme for (space-time) marked Poisson processes (free birth-and-death process, free loss networks), and (ii) a "cleaning" algorithm that trims out this process according to the interaction rules of the target process. The first step involves the perfect generation of "ancestors" of a given object, that is of predecessors 1 that may have an influence on the birth-rate under the targe...

We give a new method for generating perfectly random samples from the stationary distribution of a Markov chain. The method is related to coupling from the past (CFTP), but only runs the Markov chain forwards in time, and never restarts it at previous times in the past. The method is also related to an idea known as PASTA (Poisson arrivals see time averages) in the operations research literature. Because the new algorithm can be run using a read-once stream of randomness, we call it read-once CFTP. The memory and time requirements of read-once CFTP are on par with the requirements of the usual form of CFTP, and for a variety of applications the requirements may be noticeably less. Some perfect sampling algorithms for point processes are based on an extension of CFTP known as coupling into and from the past; for completeness, we give a read-once version of coupling into and from the past, but it remains unpractical. For these point process applications, we give an alternative...

We discuss how the ideas of producing perfect simulations based on coupling from the past for finite state space models naturally extend to multivariate distributions with infinite or uncountable state spaces such as autogamma, auto-Poisson and auto-negative-binomial models, using Gibbs sampling in combination with sandwiching methods originally introduced for perfect simulation of point processes.

Abstract not found

An overview is given of some recent work (joint with Adrian Baddeley of Perth and Colette van Lieshout of Warwick) on a new class of random point and set processes, obtained using a rather natural weighting procedure employing quermass integrals. The concept of exact (or perfect) simulation of point processes is then introduced, and a discussion is given of possibilities for perfect simulation of quermass weighted processes. 1 Introduction. This short paper is a progress report on some recent work of mine (partly in collaboration with Adrian Baddeley of Perth and Colette van Lieshout of Warwick); concerning new models for point processes and random sets, and the application to them of a new technique for "exact" or "perfect" simulation. 2 A brief overview of quermass-interaction processes In this section we review the idea of weighting point processes and random sets using quermass integrals. Recall Baddeley and Van Lieshout's definition [BvL95] of an area-interaction point process: ...

Recently Propp and Wilson [14] have proposed an algorithm, called Coupling from the Past (CFTP), which allows not only an approximate but perfect (i.e. exact) simulation of the stationary distribution of certain finite state space Markov chains. Perfect Sampling using CFTP has been successfully extended to the context of point processes, amongst other authors, by Haggstrom et al. [5]. In [5] Gibbs sampling is applied to a bivariate point process, the penetrable spheres mixture model [19]. However, in general the running time of CFTP in terms of number of transitions is not independent of the state sampled. Thus an impatient user who aborts long runs may introduce a subtle bias, the user impatience bias. Fill [3] introduced an exact sampling algorithm for finite state space Markov chains which, in contrast to CFTP, is unbiased for user impatience. Fill's algorithm is a form of rejection sampling and similar to CFTP requires sufficient monotonicity properties of the transition kernel use...

Abstract not found