Persistent random walks, variable length Markov chains and piecewise deterministic Markov processes (Q2860794)

From the authors' abstract: The aim of the paper is to consider persistent random walks \({S_t} = \sum_{n = 0}^t {{X_n}} \), whose increments are Markov chains with variable order which can be infinite. This variable memory is enlighted by a one-to-one correspondence between \(({S_t})\) and a suitable variable length Markov chain. The key fact is to consider the non-Markovian letter process \(({X_n})\) as the margin of a couple \({({X_n},{M_n})_{n \geqslant 0}}\), where \({({M_n})_{n \geqslant 0}}\) stands for the memory of the process \(({X_n})\). It is proved that, under a suitable rescaling, \(({S_n},{X_n},{M_n}))\) converges in distribution towards a time continuous process \(({S^0}(t),X(t),M(t)\). The process \({S^0}(t)\) is a semi-Markov and piecewise deterministic Markov process whose paths are piecewise linear.