SIA matrices and non-negative subdivision (Q1759049)

Although the convergence of non-negative multivariate subdivision is well characterized (see [\textit{X.-L. Zhou}, Proc. Am. Math. Soc. 134, No. 3, 859--869 (2006; Zbl 1084.65022)]), the connection to older convergence results for non-homogeneous finite Markov processes was never clearly described. In this paper, the authors discuss the uniform convergence of non-negative multivariate subdivision into this context. They use results from infinite products of stochastic matrices which appear in the theory of non-homogeneous Markov processes. A row stochastic matrix, which is stochastic, indecomposable and aperiodic, is called SIA matrix due to \textit{J. Wolfowitz} [Proc. Am. Math. Soc. 14, 733--737 (1963; Zbl 0116.35001)]. Finite families of SIA matrices and the connectivity of their directed graphs are applied to the convergence of non-negative multivariate subdivision. Finally, the uniform convergence of non-negative bivariate subdivision is considered.