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Sur les grandes déviations abstraites. Applications aux temps de séjours moyens d’un processus. (French) Zbl 0552.60024

Sémin. probabilités XVIII, 1982/83, Proc., Lect. Notes Math. 1059, 82-90 (1984).
[For the entire collection see Zbl 0527.00020.]
S. R. S. Varadhan proved [Commun. Pure Appl. Math. 19, 261-286 (1966; Zbl 0147.155)] that if a large deviation theorem with rate I holds for a family \((P_{\alpha})_{\alpha >0}\) of probability measures on a Polish space X, and if F is a bounded, continuous function on X, then \[ (*)\quad \lim_{\alpha \to \infty}\quad \alpha^{-1} \log E_{\alpha}(e^{\alpha F})=\sup_{x\in X}(F(x)-I(x)), \] where \(E_{\alpha}\) denotes expectation relative to \(P_{\alpha}\). The converse is proved here: if there exists a suitable rate (or ”action”) function I then (*) implies a large deviation result. The second part of the paper concerns the large deviation property for the ”normalized occupation time” \(L_ t(\omega,A)=t^{-1}\int^{t}_{0}1_ A(X_ s(\omega))ds\) of a general stochastic process X.
Reviewer: J.Mitro

MSC:

60F10 Large deviations
60J55 Local time and additive functionals
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