Abstract
A random walk describes the movement of a particle in discrete time, with the direction and the distance traversed in one step being governed by a probability distribution. In a correlated random walk (CRW) the movement follows a Markov chain and induces correlation in the state of the walk at various epochs. Then, the walk can be modelled as a bivariate Markov chain with the location of the particle and the direction of movement as the two variables. In such random walks, normally, the particle is not allowed to stay at one location from one step to the next. In this paper we derive explicit results for the following characteristics of the CRW when it is allowed to stay at the same location, directly from its transition probability matrix: (i) equilibrium solution and the fast passage probabilities for the CRW restricted on one side, and (ii) equilibrium solution and first passage characteristics for the CRW restricted on bath sides (i.e., with finite state space).
Highlights
A random walk describes the movement of a particle in discrete time, with the direction and the distance traversed in one step being governed by a probability
In a correlated random walk (CRW), the probability distribution at successive epochs of the walk is dependent on the preceding steps in a Markovian structure
Starting with Gillis (1955) and Mohan (1955), several authors have investigated problems related to CRW’s in one and more dimensions, and its special cases such as the symmetric CRW which has the same probability for proceeding in the same direction in successive epochs
Summary
A random walk describes the movement of a particle in discrete time, with the direction and the distance traversed in one step being governed by a probability. Starting with Gillis (1955) and Mohan (1955), several authors have investigated problems related to CRW’s in one and more dimensions, and its special cases such as the symmetric CRW which has the same probability for proceeding in the same direction in successive epochs. The results are mostly in terms of generating functions In this investigation we propose to concentrate on the equilibrium distribution of the walk when restrictions arc imposed on the state space on one or both sides and some characteristics such as the probability of fast remm into a set of contiguous states and first passage probabilities and mean passage times with respect to specified states.
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