Perfect sampling from the limit of deterministic products of stochastic matrices

Örjan Stenflo

doi:10.1214/ecp.v13-1409

Abstract

We illustrate how a technique from the theory of random iterations of functions can be used within the theory of products of matrices. Using this technique we give a simple proof of a basic theorem about the asymptotic behavior of (deterministic) "backwards products" of row-stochastic matrices and present an algorithm for perfect sampling from the limiting common row-vector (interpreted as a probability-distribution).

Highlights

Such matrices are of interest in the theory of Markov chains since if {Xn} is a Markov chain with a transition matrix, P, which is SIA in the long run Xn will be distributed according to the common row vector of Q independent of the value of X0
“forwards products”, defined by multiplying the matrices in reversed order will typically not converge. These notions have their roots in the theory of time-inhomogeneous Markov chains where “forwards/backwards products” corresponds to running the Markov chain forwards/backwards in “time”
In the particular case when n0 = 1 our algorithm reduces to a “Coupling From The Past (CFTP)”-algorithm for “perfect sampling” from the stationary distribution of a homogeneous Markov chain with a SIA transition matrix

Summary

Introduction

Such matrices are of interest in the theory of Markov chains since if {Xn} is a Markov chain with a transition matrix, P, which is SIA in the long run Xn will be distributed according to the common row vector of Q independent of the value of X0. It is not sufficient to require merely that A1, ..., An0 are SIA in Wolfowitz’ theorem since it is e.g. easy to construct two SIA matrices A1 and A2 such that their product A1A2 is decomposable and (A1A2)n will not have approximately identical rows for any n, see e.g. Hajnal [4], Equation 4(b).

Results

Conclusion