The Finite-step Random Walk With Three Absorbent Boundaries

Random walk is one of the simplest and the most important random processes in special case. The most existing literature related to the random walks are involved in infinite steps and based on the Law of Large Numbers. However, in many real problems, especially in the fields of economics, biology and gambling games, we have to deal with the unsymmetrical and finite step random walk with absorbent boundaries. In this work, we have studied the model of random walk with three absorbent boundaries. Firstly we give the computing formulae of the probabilities that the random point at original point is absorbed by the three absorbent boundaries in one or two pace random walk. Secondly, we achieve the average time that the random point reaches the three absorbent boundaries respectively. Finally, we discuss the problems that the probabilities change with the changes of the several parameters in the formulae and explore several properties in the process of random walking.