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JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range of content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms of scholarship. For more information about JSTOR, please contact support@jstor.org. Institute of Mathematical Statistics is collaborating with JSTOR to digitize, preserve and extend access to The Annals of Probability. A Gaussian upper bound for the iterated kernels of Markov chains is obtained under some natural conditions. This result applies in particular to simple random walks on any locally compact unimodular group G which is compactly generated. Moreover, if G has polynomial volume growth, the Gaussian upper bound can be complemented with a similar lower bound. Various applications are presented. In the process, we offer a new proof of Varopoulos' results relating the uniform decay of convolution powers to the volume growth of G. 1. Introduction. The first result proved in this paper is a fairly general Gaussian upper bound for Markov chains. This bound applies in particular to simple random walks on locally compact compactly generated unimodular groups. When the group has polynomial volume growth, the iterated convolu-tion kernel governing the random walk is shown to satisfy a two sided Gaussian estimate. Various applications of this estimate are discussed. Our Gaussian upper bound for Markov chains is as follows. Consider a symmetric Markov kernel k defined on a measure space X, and assume that there is a distance function p on X such that k(x, y) = 0 whenever x, y satisfy p(x, y) ? ro. Also assume that the iterated kernels kn satisfy the uniform estimate

Gaussian Estimates for Markov Chains and Random Walks on Groups