The Annals of Mathematical Statistics

A Potential Theory for Supermartingales

E. Count Curtis

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A potential theory for supermartingales is presented below. It is much like the classical Newtonian potential theory and is a generalization of the potential theory for transient Markov chains. While dealing with stochastic processes with more general dependence relations the new theory retains what we believe to be the important features of the transient Markov chain theory. Briefly, a pure potential is a non-negative supermartingale $\{Z_n, F_n\}$ which satisfies the condition $E\lbrack Z_{n+k}\mid F_n\rbrack \rightarrow 0 \text{a.e. as} k \rightarrow \infty$ for every $n.$ The potential principles of domination, Riesz decomposition, lower envelope, balayage, equilibrium and minimum are proved for these potentials. It is shown how the corresponding results of the transient Markov chain theory can be derived from the new theory. Also, some applications to standard martingale theory are given.

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Ann. Math. Statist., Volume 39, Number 3 (1968), 802-814.

First available in Project Euclid: 27 April 2007

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Curtis, E. Count. A Potential Theory for Supermartingales. Ann. Math. Statist. 39 (1968), no. 3, 802--814. doi:10.1214/aoms/1177698312.

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