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In an influential paper, Courtois and Semal (1984) establish that when $G$ isan irreducible substochastic matrix for which $\sum_{n=0}^{\infty}G^n <\infty$,then the stationary distribution of any stochastic matrix $P\ge G$ can beexpressed as a convex combination of the normalized rows of $(I-G)^{-1} =\sum_{n=0}^{\infty} G^n$. In this note, we give a short proof of this resultthat extends the theory to the countably infinite and continuous state spacesettings. This result plays an important role in obtaining error bounds inalgorithms involving nearly decomposable Markov chains, and also in statetruncations for Markov chains. We also use the representation to establish anew total variation distance error bound for truncated Markov chains.

A Short Proof of a Convex Representation for Stationary Distributions of Markov Chains with an Application to State Space Truncation