Symmetric distributions of random measures in higher dimensions
An infinite sequence of random variables \(X= (X_1,X_2, \dots)\) is said to be spreadable if all subsequences of \(X\) have the same distribution. Ryll-Nardzewski showed that \(X\) is spreadable if and only if \(X\) is exchangeable. The main result shows that a random measure on the space \({\mathcal W}_d =\{(x_1, \dots, x_d)\in \mathbb{R}^d_+: x_1<x_2 <\cdots < x_d\}\) is spreadable if and only if it can be extended to an exchangeable random measure on \(\mathbb{R}^d_+\). This result is an extension of a result of \textit{O. Kallenberg} [J. Theor. Probab. 5, No. 4, 727-765 (1992; Zbl 0759.60035)] to a continuous parameter setting.
scientific article; zbMATH DE number 1061819