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Geometric constructions with discretized random variables. (English) Zbl 1097.65018
A multivariate generalization of thicket construction is considered. A thicket in \(\mathbb{R}^d\) is a discretized representation of a random vector distribution in the form of finite set \(T=\{(C_i,p_i)\}\), where \(C_i\) are open convex sets in \(\mathbb{R}^d\) and \(p_i\) are probabilities with \(\sum p_i=1\). Lower \(\underline{p}\) and upper \(\overline{p}\) probabilities for \(T\) are defined by \[ \underline{p}(X)=\sum_{C_i\subset X}p_i\;\;\;\overline{p}(X)=\sum_{C_i\cap X\not=\emptyset} p_i. \] A random variable \(\xi\) is represented by a thicket \(T\) if \(\underline{p}(X)\leq \mathbf{P}\{\xi\in X\}\leq\overline{p}(X)\). Algorithms for constructing a thicket with given lower and upper probabilities are described. The authors introduce the concept of nested thicket representation for r.v.s which minimize the information loss. An algorithm is described for construction of a thicket \(T\) which represents \(\xi *\eta\) if representations for \(\xi\) and \(\eta\) are given. Here \(*\) is any binary operation.
MSC:
65C50 Other computational problems in probability (MSC2010)
60E05 Probability distributions: general theory
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