An extension of the contraction principle
The ``contraction principle'' is about large deviations; the basic definitions figure in [\textit{A. Dembo} and \textit{O. Zeitouni}, ``Large deviation techniques and applications''. 2nd ed. (1998; Zbl 0896.60013)]. Its statement is (C). Let \((X_{n})\) satisfy the large deviation principle with good rate function \(I\) (defined on the metric space \({\mathcal X}\)), \(G:{\mathcal X}\rightarrow {\mathcal Y}\), \({\mathcal Y}\) metric. Suppose that, for every \(x\) with \(I(x)<\infty\): for every sequence \(x_{n}\rightarrow x\), \(\lim_{k}\, G(x_{n_{k}})\) exists for some subsequence and, for every partial limit \(y\) of \(G\) in \(x\) (situation denoted \(y\in G^{x}\)), there exists a sequence \(x'_{n}\rightarrow x\) such that \(G(x'_{n})\rightarrow y\), \(I(x'_{n})\rightarrow I(x)\) and \(G\) is continuous at \(x'_{n}\). Then, defining \(I_{0}(x,y)=I(x)\) when \(y\in G^{x}\) and \(\infty\) otherwise, \(((X_{n},G(X_{n})))\) satisfies the large deviation principle with good rate function \(I_{0}\). The author proves a ``contraction principle'' in which \({\mathcal X}\), \({\mathcal Y}\) are topological spaces, in the first condition a net appears instead of the sequence \(x_{n}\) (and a subnet instead of \(n_{k}\); the condition on \(G\) is named ``almost compactness in \(x\)''), but with \(I\), \(I_{0}\) not necessarily good. This result is preceded by ones about lower and upper bound principles, respectively, proved in more general conditions, all in terms of \(G,I\). The fact that \(I_{0}\) is good is equivalent, when \(I\) is locally bounded on \((I<\infty )\), to: \(I\) good and \(G\) is almost compact in all \(x\). The author proves then that the exponential tightness for \((X_{n})\) implies the same for \((G(X_{n}))\), if \(G:{\mathcal X}\rightarrow {\mathcal Y}\), \({\mathcal Y}\) metric and \(G\) is almost compact at all \(x\). As an application of (C): \(((X_{n},Y_{n}))\) (instead of \((X_{n})\)), \(X_{n}\), \(Y_{n}\) have values in the set of all continuous \(f:[0,\infty )\rightarrow {\mathbb R}^{d}\), \(f(0)= 0\), \(I(x,y)=\int (\| x'(t)\|^{2}+\| y'(t)\|^{2})dt\), \(G(x,y)=(g(x_{j},y_{j}))_{j=1,...,d}\), \(g(u,v)(t)= \sum u(\tau_{i-1})(v(\min(\tau_{i},t))-v(\min(\tau_{i-1},t)))\), \(\tau_{0}=0\), \(\tau_{i}= \min\{t>\tau_{i-1},| u(t)-u(\tau_{i-1})| >\varepsilon\}\) for some fixed \(\varepsilon > 0\); it is proved that (C) applies. The paper concludes with a review of different variants of the contraction principle and has a paragraph with proofs of several facts about quasicontinuous functions.
scientific article; zbMATH DE number 2083454