Exponential tightness can fail in the strong topology
The authors point out that a large deviations principle does not imply the exponential tightness in general. In Theorem 1.1 they prove for the classical Sanov result that the exponential tightness holds in the strong topology if and only if the underlying measure is purely atomic. In some remarks exercises or results by several authors are cited [see \textit{P. Gänssler}, Z. Wahrscheinlichkeitstheorie Verw. Geb. 17, 124-146 (1971; Zbl 0195.33902); \textit{J. Lynch} and \textit{J. Sethuraman}, Ann. Probab. 15, 610-627 (1987; Zbl 0624.60045); \textit{A. Dembo} and \textit{O. Zeitouni}, ``Large deviations techniques and applications'' (1993; Zbl 0793.60030); \textit{A. de Acosta}, in: Studies in applied probability. Essays in honour of Lajos Takács, 41-47 (1994; Zbl 0810.60022); \textit{D. W. Stroock}, ``Probability theory: an analytic view'' (Cambridge, 1993)].
scientific article; zbMATH DE number 1348355