Random Chains and Systems with Complete Connections

1 Random Chains and Systems with Complete Connections 

If we consider the example described at the beginning of this chapter, the element of study in stochastic modelling is the particle which moves in a trajectory where the local state of displacement is randomly chosen. The description for this discrete displacement and its associated general model, takes into consideration the 

The process described above is thus repeated with constant time intervals. So, we have a discrete time t = nAr where n is the number of displacement steps. By the rules of probability balance and by the prescriptions of the Markov chain theory, the probability that shows a particle in position i after n motion steps and having a k-type motion is written as follows  

In order to begin the calculations, we need to know some parameters such as the process components (k = 2 or k = 3, etc.), the trajectory matrix (p in the model), and the equation that describes the distribution function of the path length for displacement k and for the initial state of the process 

i — a), Vi Z and k K. In our example, when the particle displacement is realized by unitary steps and in a positive direction (type I) or in a negative direction (type II) and where the path length distribution is uniform with a unitary value for both component processes, we have  

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