Randomized trial explores weak convergence of probability measures in relation to energy functions and manifolds, suggesting new insights.
Let Q be a fixed probability on the Borel σ-field in Rⁿ and H be an energy function continuous in Rⁿ. A set N is related to H by N = infyH(y) = H(x)\. Laplace's method, which is interpreted as weak convergence of probabilities, is used to introduce a probability P on N. The general properties of P are studied. When N is a union of smooth compact manifolds and H satisfies some smooth conditions, P can be written in terms of the intrinsic measures on the highest dimensional mainfolds in N.
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Chii-Ruey Hwang (1980) studied this question.
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