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October, 1972 Convergence in Distribution of Random Measures
Miloslav Jirina
Ann. Math. Statist. 43(5): 1727-1731 (October, 1972). DOI: 10.1214/aoms/1177692410


Let $(S, \mathscr{I})$ be a measurable space, $M$ the set of all finite measures on $\mathscr{I}, \mathscr{F}_M$ the $\sigma$-algebra generated by the family of all measurable cylindrical sets $\cap^k_{i=1} \{\mu \in M: \mu(A_i) \leqq a_i\}$. With each probability measure $P$ on $\mathscr{F}_M$ the family $\{P_{A_1}, \cdots, A_k\}$ of all finite-dimensional probability measures of the cylidrical sets is associated. The following problem is considered: Given a sequence $P^{(n)}$ of probability measures on $\mathscr{F}_M$ such that each sequence $P^{(n)}_{A_1}, \cdots, A_k$ converges weakly to a $k$-dimensional probability measure $P_{A_1}, \cdots, A_k$, does the family $\{P_{A_1}, \cdots, A_k\}$ generate a probability measure $P$ on $\mathscr{F}_M?$ It is proved that the answer is affirmative if $(S, \mathscr{I})$ is the Euclidean $n$-space with the $\sigma$-algebra of Borel sets.


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Miloslav Jirina. "Convergence in Distribution of Random Measures." Ann. Math. Statist. 43 (5) 1727 - 1731, October, 1972.


Published: October, 1972
First available in Project Euclid: 27 April 2007

zbMATH: 0249.60003
MathSciNet: MR362426
Digital Object Identifier: 10.1214/aoms/1177692410

Rights: Copyright © 1972 Institute of Mathematical Statistics


Vol.43 • No. 5 • October, 1972
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