$\boldsymbol{\mathcal{G}-}$Inhomogeneous Markov Systems of High Order

In the present, we introduce and study the G-\mathcal{G-}inhomogeneous Markov system of high order, which is a more general in many respects stochastic process than the known inhomogeneous Markov system. We define the inhomogeneous superficial razor cut mixture transition distribution model extending for the homogeneous case the idea of the mixture transition model. With the introduction of the appropriate vector stochastic process and the establishment of relationships among them, we study the asymptotic behaviour of the G-\mathcal{G-}inhomogeneous Markov system of high order. In the form of two theorems, the asymptotic behaviour of the inherent G-\mathcal{G-}inhomogeneous Markov chain and the expected and relative expected population structure of the G-\mathcal{G-}inhomogeneous Markov system of high order, are provided under assumptions easily met in practice. Finally, we provide an illustration of the present results in a manpower system.