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We consider a finite random walk on a weighted graph G; we show that the sample average of visits to a set of vertices A converges to the stationary probability /spl pi/(A) with error probability exponentially small in the length of the random walk and the square of the size of the deviation from /spl pi/(A). The exponential bound is in terms of the expansion of G and improves previous results. We show that the method of taking the sample average from one trajectory is a more efficient estimate of /spl pi/(A) than the standard method of generating independent sample points from several trajectories. Using this more efficient sampling method, we improve the algorithms of Jerrum and Sinclair (1989) for approximating the number of perfect matchings in a dense graph and for approximating the partition function of an Ising system. We also give a fast estimate of the entropy of a random walk on an unweighted graph.