Classical and Modern Branching Processes

Volume 84 of the series The IMA Volumes in Mathematics and its Applications pp 305-321

Markov Cascades

  • Edward C. WaymireAffiliated withDepartment of Mathematics, Oregon State University
  • , Stanley C. WilliamsAffiliated withDepartment of Mathematics, Utah State University

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The Kahane-Peyrière theory of homogeneous independent cascades is extended to a class of finite state Markov dependent cascades. The problems of (i) non-triviality (ii) divergence of moments and (iii) carrying dimension are completely solved for this class of models. While the authors have obtained complete solutions to problems (i) and (iii), and partial solutions to (ii) in wider generality, the focus of the present paper is on fixed points of a naturally associated random recursive equation of independent interest in diverse contexts. By essentially algebraic methods of this paper we obtain complete solutions to all three problems (i)-(iii). In particular an interesting role of a “mean-reversal” symmetry in the computation of critical parameters for survival and carrying dimension is uncovered.