Transient limits

Let $mathbf{A}$ be a {sc Markov} matrix depending on a small parameter $sigma$, and $mathbf{C}_{n}$ the average of the first $n$ powers of $mathbf{A}$. The stationary distributions of $mathbf{A}$ are the rows of $mathbf{S}%=limlimits_{nightarrow +infty}mathbf{C}_{n}$. The extit{limiting stationary distributions} are the rows of $limlimits_{sigmaightarrow 0}mathbf{S}$. We investigate extit{transient limits} of the sequence $mathbf{C}_{n}$. These idempotent {sc Markov} matrices come up implicitly in an algorithm to compute limiting stationary distributions. They represent the intermediate-term behavior of the {sc Markov} chain at different time scales.