On the solutions and the steady states of a master equation

A complete characterization of the time behavior of the means and variance of a stochastic process which is generated by a finite number of independent systems is presented based on the master equation for the conditional probability. It is found that the means and variance relax to a steady state and that the steady state will be independent of the initial state if and only if a matrix related to the transition matrix is nonsingular. Finally, the result that the variance approaches its steady-state form at twice the rate of the means is shown to depend on the nonsingularity of the same matrix.