This paper is concerned with approximating the leading components of the stationary vector of a semi-infinite discrete markov chain. The most widely treated method extracts a leading principal submatrix from the matrix of transition probabilities, adjusts its elements so that it becomes stochastic, and takes the stationary vector of the result as the approximation. In this paper, the consequences of taking the normalized Perron vector of the unadjusted matrix as the approximation are explored. Error bounds are derived, and it is shown that the adjusted and unadjusted methods are approximations to one another. (UMIACS-TR-2003-65)