Abstract
A sufficient condition for a sequence of random variables, $T_1, T_2,\cdots$, with cumulant generating functions, $\psi_1, \psi_2,\cdots$, to have a large deviation rate is that $n^{-1}\psi_n(\lambda)\rightarrow \psi(\lambda)$, where $\psi(\lambda)$ satisfies certain regularity conditions. Here it is shown that, when the large deviation rate exists and $T_1, T_2,\cdots$ are properly truncated, it is a necessary condition.
Citation
James Lynch. "A Curious Converse of Siever's Theorem." Ann. Probab. 6 (1) 169 - 173, February, 1978. https://doi.org/10.1214/aop/1176995623
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