A Markov chain whose state space is given by the integers i = 0, ±1, ±2, ... is said to be a random walk if, for some number 0 <>p 1, Pi,i+1 = p = 1 - Pi,i-1, i = 0, ±1, ...
The preceding Markov chain is called a random walk for we may think of it as being a model for an individual walking on a straight line who at each point of time either takes one step to the right with probability p or one step to the left with probability 1 - p.