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By Daniel J. Velleman

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Extra resources for American Mathematical Monthly, volume 117, number 3, March 2010

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19] Thus, in fact, P´olya’s motivation was to understand the number of collisions C of two independent random walks. For random walks on a lattice, this is equivalent to March 2010] ´ POLYA’S THEOREM VIA HIS URN 229 considering returns to zero for a new random walk. On a more general graph, however, the three properties (i)–(iii) on page 221 are not equivalent if C replaces Na . It is true that for recurrent graphs of bounded degree, EC = ∞; nevertheless, this does not imply that P{C = ∞} = 1.

Let Z n = {v : d(a, v) ≥ n}, and create a finite graph Gn by identifying all vertices in Z n with a new vertex z n , and removing all edges with both vertices in Z n . Let In be the unit current flow on Gn from a to z n . If Nn (x, y) is the net number of traversals of the oriented edge (x, y) in Gn before hitting z n , then In (x, y) = Ea Nn (x, y). With a minor abuse of notation, Nn (x, y) is also the net number of traversals of the oriented edge (x, y) in G before hitting Z n ; with this interpretation, Nn (x, y) ↑ N (x, y), where N (x, y) is the net number of traversals of (x, y) for the random walk in G started at a and run for infinite time.

His research interests are in probability theory, and he is a co-author of the book Markov Chains and Mixing Times (AMS, 2008). D. Hebrew University, Jerusalem, 1990) is a principal researcher and manager of the Theory Group at Microsoft Research in Redmond. He is also an adjunct professor at the University of California, Berkeley and at the University of Washington, Seattle. He is the author of more than 150 research papers and co-author of the book Markov Chains and Mixing Times (AMS, 2008). His research interests include most areas of probability theory, as well as parts of ergodic theory, game theory, and information theory.

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American Mathematical Monthly, volume 117, number 3, March 2010 by Daniel J. Velleman


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