Correlation Polytopes and the Geometry of Limit Laws in Probability
Let be n events in a probability space, and suppose that we have only partial information about the distribution: The probabilites of the events themselves, and their pair intersections. With this partial information we cannot, usually, deternine the probability of an event B in the algebra generated by the 's, but we can obtain lower and upper bounds. This is done by a linear program related to the correlation polytope c(n), a structure introduced in , . In the first part of the paper I demonstrate how laws of large numbers (for sequences of events which are not necessarily independent) can be proved, using only the duality theorem of linear programming. These include the weak law of large numbers (necessary and sufficient condition) and various sufficient conditions for strong laws. The connection between these laws and the facet structure of the correlation polytope is established. In the second part of the paper I consider a more general case. Assume that our information consists of the values of the probabilities of all intersections of the 's up to size k, k < n. The techniques of linear programming lead naturally to an application of the theory of polynomial approximation in estimating the size of various events. In particular, I prove an approximate version of the central limit theorem.
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