Examples of Markov Chains - A Center-biased Random Walk

A Center-biased Random Walk

Consider a random walk on the number line where, at each step, the position (call it x) may change by +1 (to the right) or -1 (to the left) with probabilities:

(where c is a constant greater than 0)

For example if the constant, c, equals 1, the probabilities of a move to the left at positions x = -2,-1,0,1,2 are given by respectively. The random walk has a centering effect that weakens as c increases.

Since the probabilities depend only on the current position (value of x) and not on any prior positions, this biased random walk satisfies the definition of a Markov chain.