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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.

##### MSC:
 60F10 Large deviations