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Capacities, large deviations and log log laws. (English) Zbl 0724.60033
Stable processes and related topics, Sel. Pap. Workshop, Ithaca/NY (USA) 1990, Prog. Probab. 25, 43-83 (1991).
[For the entire collection see Zbl 0718.00011.]
The space \({\mathcal C}(E)\) of all (not necessarily Choquet) capacities on a separable metric space E is endowed with the vague and narrow topologies and corresponding Borel fields. A net (or sequence) \(c_ n\to c\) in \({\mathcal C}(E)\) vaguely (narrowly) if lim inf \(c_ n(G)\geq c(G)\) for open \(G\subset E\), lim sup \(c_ n(B)\leq c(B)\) for compact (closed) \(B\subset E\). One can consider random measures, point processes, extremal processes, upper semicontinuous processes and random closed sets as the families of random variables with values in natural subspaces of \({\mathcal C}(E)\). Large deviation principles (LDP) are viewed as a class of limit relations of the type \(c_ n^{1/a_ n}\to c\) (vaguely or narrowly) for capacities \(c,c_ n\), \(n\in {\mathbb{N}}\), and a sequence \(a_ n>0\) with lim \(a_ n=\infty\). After a “change of time” narrow LDP, where \(c_ n\) are probability distributions of r.v. \(X_ n\) in E on a probability space, can be transformed into a pre log log law \(P(X_{n(m)}\in \circ)=m^{-J(\cdot)+o(1)}\) narrowly with \(c=e^{-J}\), J an “inf measure”. The log log law meaning that \((X_{n(m)})\) is w.p. 1 relatively compact in E with the set of limit points \(\{x\in E:\;J(\{x\})\leq 1\}\) holds for sufficiently independent \(X_{n(m)}\). With E being the space of capacities on a positive quadrant \({\mathbb{R}}^ 2_+\), specific LDP and log log laws are formulated and discussed for the E- valued Poisson process.

60F10 Large deviations