Multimode time-Markov systems: Recursive tensor-based analysis, chaotic generation, locally looping processes

The paper proposes a systematic solution to the problem of mixing different stochastic processes, each implied by a certain mode of operation of the system at hand and with a random duration whose distribution depends on the previous and present modes. We do so by widening the scope of an existing framework for the statistical characterization of finite valued processes with memory-one properties. The point of view is that of stochastic dynamics and the state space of the process is partitioned into regions (that we identify with modes) such that, if sojourn in a mode can be assumed, the statistical characterization is fully understood. The process is also allowed to stochastically move from one mode to another and the number of time steps for which it remains in each mode is a random variable whose distribution is a function only of the mode visited before. A general theoretical framework is developed here for the computation of any-order joint probabilities. The framework is then exemplified for the case of locally looping systems that are random sequences of modes comprising the cyclical execution of given atomic actions. They are the model of choice for complex appliances that operate following the steps of a communication protocol, and/or the various phases of a bus cycle, and/or the load-computestore mechanism of a microprocessor, etc. Exploiting the theory put forward by the paper, we highlight how these processes could be generated by suitably designed 2-d chaotic maps and how their second- and third-order spectra may be obtained and interpreted when exponentially or polynomially decaying distributions are assumed for mode sojourn times.