Inhomogeneous Products of Non-negative Matrices

Abstract

In a number of important applications the asymptotic behaviour as r → ∞ of one of the

EquationSource% MathType!MTEF!2!1!+- % feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqabeaacaWGgb % Gaam4BaiaadkhacaWG3bGaamyyaiaadkhacaWGKbGaciiuaiaackha % caWGVbGaamizaiaadwhacaWGJbGaamiDaiaadohacaGG6aGaamivam % aaBaaaleaacaWGWbGaaiilaiaadkhaaeqaaOGaeyypa0ZaaiWaaeaa % caWG0bWaa0baaSqaaiaadMgacaWGQbaabaWaaeWaaeaacaWGWbGaai % ilaiaadkhaaiaawIcacaGLPaaaaaaakiaawUhacaGL9baacqGH9aqp % caWGibWaaSbaaSqaaiaadchacqGHRaWkcaaIXaaabeaakiaadIeada % WgaaWcbaGaamiCaiabgUcaRiaaikdaaeqaaOGaeyyXICTaeyyXICTa % eyyXICTaamisamaaBaaaleaacaWGWbGaey4kaSIaamOCaaqabaaake % aacaWGcbGaamyyaiaadogacaWGRbGaam4DaiaadggacaWGYbGaamiz % aiGaccfacaGGYbGaam4BaiaadsgacaWG1bGaam4yaiaadshacaWGZb % GaaiOoaiaadwfadaWgaaWcbaGaamiCaiaacYcacaWGYbaabeaakiab % g2da9maacmaabaGaamyvamaaDaaaleaacaWGPbGaamOAaaqaamaabm % aabaGaamiCaiaacYcacaWGYbaacaGLOaGaayzkaaaaaaGccaGL7bGa % ayzFaaGaeyypa0JaamisamaaBaaaleaacaWGWbGaey4kaSIaamOCaa % qabaGccaWGibWaaSbaaSqaaiaadchacqGHRaWkcaaIYaaabeaakiab % gwSixlabgwSixlabgwSixlaadIeadaWgaaWcbaGaamiCaiabgUcaRi % aaigdaaeqaaaaaaa!95EE!]]</EquationSource><EquationSource Format="TEX"><![CDATA[$$ \begin{gathered} Forward\Pr oducts:{T_{p,r}} = \left\{ {t_{ij}^{\left( {p,r} \right)}} \right\} = {H_{p + 1}}{H_{p + 2}} \cdot \cdot \cdot {H_{p + r}} \hfill \\ Backward\Pr oducts:{U_{p,r}} = \left\{ {U_{ij}^{\left( {p,r} \right)}} \right\} = {H_{p + r}}{H_{p + 2}} \cdot \cdot \cdot {H_{p + 1}} \hfill \\ \end{gathered} $$

and its dependence on p is of interest, where {H _k ,k = 1, 2, ...} is a set of (n × n) matrices satisfying H _k ≥0. We shall write H _k ={h _ij (k)}, i, j=1, ...;, n. The kinds of asymptotic behaviour of interest are weak ergodicity and strong ergodicity, and a commonly used tool is a contraction coefficient (coefficient of ergodicity). We shall develop the general theory in this chapter. The topic of inhomogeneous products of (row) stochastic matrices has special features, and is for the most part deferred to Chapter 4.