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pagerank - The Stationary Distribution of a Markov Chain...

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The Stationary Distribution of a Markov Chain Alessandro Panconesi DI, La Sapienza of Rome May 15, 2005 In this note I present a concise proof of the existence and uniqueness of the limit distribution of an ergodic markov chain. This nice proof was described to me by David Gilat of Hebrew University during a hot summer afternoon in Perugia. A finite n × n transition probability matrix P := [ p ij ] is a stochastic matrix where p ij is the transition probability of going from state i to state j . We can think of a directed graph whose edges are weighted with transition probabilities. Note that since P is stochastic, the sum of the probabilities of the arcs outgoing from a vertex i sum up to 1, i.e. j p ij = 1. We are interested in keeping track of the random walk of a pebble. The pebble is initially placed on the graph according to an initial distribution X 0 and then it proceeds by traversing the edges ij ’s with their associated probabilities p ij ’s. X 0 is a probability distribution, X i 0 being the probability of placing the pebble on vertex i initially. After one step the position of the pebble is given by the probability distribution X 1 = X 0 P and in general after t steps we have X t = X t - 1 P = X 0 P t . The infinite sequence X 0 , X 1 , X 2 , . . . is called a markov chain . In what follows we shall use the term markov chain somewhat loosely, sometimes referring to the sequence proper, sometimes to the transition matrix P or the underlying graph. The meaning will be clear from the context.

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