
⋮ 
Thus, for a family starting in the middle economic class, after 10 generations the probability that the family will be in the upper class is .2, the middle class .553, and the lower class .25.
Now consider the initial state vector corresponding to starting in the lower class:
Then we have for x^{(10)}:
x^{(10)} = T^{10}x^{(0)} = 




Thus, for a family starting in the lower economic class, after 10 generations the probability that the family will be in the upper class is .2, the middle class .547, and the lower class .255.
Notice that the system is converging to a certain set of probabilities for social mobility as the number of generations increases. Also, the set of probabilities that the system is converging to appears to be independent of the initial class of the family. We will now formalize these observations.
Regular Markov Processes and Equilibrium
A Markov process is called a regular Markov process if some power of the corresponding transition matrix has only positive elements. It can be shown that every regular Markov process eventually converges to a certain state vector, called the steadystate vector, that is independent of the initial state vector. When a system reaches the steadystate vector, it is considered to be an equilibrium. The equilibrium of Markov processes is an appealing quality of certain real life models, such as population statistics after multiple generations. [2]
Consider the problem of trying to determine the steadystate vector s for a given regular Markov process. We know that Ts = s, since if this is not the case, then the steadystate vector is moving away from the steadystate vector, which is a contradiction. Thus, 1 is an eigenvalue of T with corresponding eigenvector s. Hence, if a Markov process is regular, we can determine the steadystate vector of the process by considering the eigenvector of T corresponding to eigenvalue 1 that is normalized such that the sum of the probabilities is equal to 1 [2]. Note that eigenvectors of T corresponding to eigenvalue 1 are unique up to common multiples if T represents a regular Markov process, so there is only one such normalized eigenvector [4].
Continuing with the above model of social mobility, we have:
