Symmetries on random arrays and set-indexed processes
The author considers processes \(X=(X_ J)\) indexed by the collection \(\tilde N\) of all finite subsets \(J=\{j_ 1,\dots,j_ d\}\) of \(N\) with \(j_ 1<\cdots<j_ d\), and processes \(Y=(Y_ s)\) indexed by the class \(\overline N\) of all finite sequences \(s=(s_ 1,\dots,s_ d)\) of distinct natural numbers. For \(J=\{j_ 1,\dots,j_ d\}\in\tilde N\), \(s=(s_ 1,\dots,s_ d)\in\overline N\) and \(I=\{i_ 1,\dots,i_ k\}\in\tilde N\cap\{1,\dots,d\}\), write \(J\circ I=\{j_{i_ 1},\dots,j_{i_ k}\}\) and \(sI=\{s_{i_ 1},\dots,s_{i_ k}\}\). For any process \(U=(U_ J; J\in\tilde N)\), define new processes \(\tilde U_ J=(U_{J\circ I}; I\subset\{1,\dots,| J|\})\), \(J\in\tilde N\), and \(\tilde U_ s=(U_{sI}; I\subset\{1,\dots,| s|\})\), \(s\in\overline N\). Further, \(X\) is said to be spreadable if \((X_{\{p_ j;j\in J\}}; J\in\tilde N){\buildrel{\text{d}}\over =}(X_ J; J\in\tilde N)\) for all subsequences \((p_ 1,p_ 2,\dots)\) of \(N\). This rather bushy paper contains essentially the following results. Theorem 1. A process \(X=(X_ J; J\in\tilde N)\) is spreadable iff \(X_ J=f(\tilde U_ J)\) a.s., \(J\in\tilde N\), for some family \(U=(U_ J; J\in\tilde N)\) of i.i.d. r.v.'s uniformly distributed on [0,1] and some measurable function \(f:\bigcup_{\alpha\geq 0}[0,1]^{2^ d}\to R\). Theorem 2. A process \(\overline Y=(Y_ s; s\in\overline N)\) is exchangeable iff \(X_ s=f(\tilde U_ s)\) a.s., \(s\in\overline N\), with \(U\) and \(f\) as in Theorem 1. A consequence of Theorems 1 and 2 is that a process \(X\) indexed by \(N\) is spreadable iff it can be extended to an exchangeable process \(Y\) indexed by \(N\).
scientific article; zbMATH DE number 96939
scientific article