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An introduction to the theory of large deviations. (English) Zbl 0552.60022
Universitext. New York etc.: Springer-Verlag. VII, 196 p. DM. 56.00; $ 20.40 (1984).
Let E be a Polish space an \(\{\mu_{\epsilon}\), \(\epsilon >0\}\) be a family of probability measures on E such that \(\mu_{\epsilon}\) converges weakly to \(\delta_{x_ o}\), as \(\epsilon\) \(\downarrow 0\). The problem is investigated how fast \(\mu_{\epsilon}(\Gamma)\to 0\) for \(\Gamma\in {\mathcal B}_ E\) and \(x_ 0\not\in {\bar \Gamma}\). The main part of the theorems proved is of the form: \[ \underline{\lim_{}\epsilon \downarrow 0}\epsilon \log \mu_{\epsilon}(G)\geq -\inf_{y\in G}I(y)\quad for\quad all\quad open\quad sets\quad G\subset E\quad and \] \[ \overline{\lim}_{\epsilon \downarrow 0}\quad \epsilon \log \mu_{\epsilon}(F)\leq -\inf_{y\in F}I(y)\quad for\quad all\quad closed\quad sets\quad F\subset E. \] Here I is the so-called rate function. The results and methods presented in the book belong to M. Shilder, R. Azencott, M. Kac, M. Donsker and S. R. S. Varadhan, D. Stroock, and other authors. The bibliography contains 6 positions.
Reviewer: A.Plikusas

60F10 Large deviations
60B10 Convergence of probability measures
60-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to probability theory
rate function