Random Walks

This chapter deals with random walk models. These models are discrete approximations of physical processes that describe the motion of diffusing particles. In this chapter, some variations of the simple random walk on the real line are described and analyzed. The simple random walk model is described and formulated as a Markov chain in Sect. 1. Section 2 deals with (i) symmetric random walk on the set of integers, (ii) random walk on the set of whole numbers with absorbing, reflecting and elastic barriers at 0. In each case, relevant results are stated, proved and illustrated. In Sect. 3, all variations of finite state space random walks with absorbing, reflecting and elastic barriers are discussed in detail. An important application of a random walk model is to the gambler’s ruin problem. In Sect. 4, this problem is described and answers to standard problems such as gambler’s ruin and duration of the game in the two cases, when total capital of the game is finite as well as infinite, are discussed. In Sect. 5, two Markov chains, namely, Ehrenfest chain and Birth-Death chain are considered. These two chains are examples of time homogeneous Markov chains with state-dependent transition probabilities and can be viewed as an extension of random walk models discussed in earlier sections. Relevant R codes are given in Sect. 6.