Question

Show that the Ehrenfest chain does indeed fulfill the Markov property.

Answer 1

Ehrensfest model is defined as supposing we have two urns, labeled 0 and 1, that contain a total of m balls. We randomly choose a ball from any of the two urns and put it into the other urn. The state of the system at time n is the number of balls in urn 1, which we will denote by Xn. Our stochastic process is X = ( X0, X1, X2, ...) with state-space S = {0, 1, ..., m}.
Then the probability that X_n+1=j given X₀, X₁, X₂, ...X_n is given by:

, but note that j can only be i-1 or i+1, i.e. the transition probabilities are given by:

. Thus as the probability of X_n+1=j depends only on X_n=i. Thus the above chain is a Markov chain.