Probability Surveys

A survey of random processes with reinforcement

Robin Pemantle

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The models surveyed include generalized Pólya urns, reinforced random walks, interacting urn models, and continuous reinforced processes. Emphasis is on methods and results, with sketches provided of some proofs. Applications are discussed in statistics, biology, economics and a number of other areas.

Article information

Probab. Surveys, Volume 4 (2007), 1-79.

First available in Project Euclid: 23 February 2007

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60J20: Applications of Markov chains and discrete-time Markov processes on general state spaces (social mobility, learning theory, industrial processes, etc.) [See also 90B30, 91D10, 91D35, 91E40] 60G50: Sums of independent random variables; random walks
Secondary: 37A50: Relations with probability theory and stochastic processes [See also 60Fxx and 60G10]

urn model urn scheme Pólya’s urn stochastic approximation dynamical system exchangeability Lyapunov function reinforced random walk ERRW VRRW learning agent-based model evolutionary game theory self-avoiding walk


Pemantle, Robin. A survey of random processes with reinforcement. Probab. Surveys 4 (2007), 1--79. doi:10.1214/07-PS094.

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