For sequences of non-lattice weakly dependent random variables, we obtain asymptotic expansions for Large Deviation Principles. These expansions, commonly referred to as strong large deviation results, are in the spirit of Edgeworth Expansions for the Central Limit Theorem. We show that Diophantine iid sequences, finite state Markov chains, strongly ergodic Markov chains and observations from smooth expanding maps satisfy the strong large deviation results. In addition, we obtain equivalent expansions in the case of stochastic processes, and verify their existence for additive functionals of processes generated from SDE's satisfying the H\"ormander condition.

中文翻译：

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大偏差的高阶渐近线——第一部分

对于非晶格弱相关随机变量序列，我们获得大偏差原理的渐近展开式。这些扩展，通常称为强大偏差结果，符合 Edgeworth Expansions for the Central Limit Theorem 的精神。我们表明丢番图 iid 序列、有限状态马尔可夫链、强遍历马尔可夫链和平滑扩展图的观察满足强大偏差结果。此外，我们在随机过程的情况下获得了等效扩展，并验证了它们的存在性，用于满足 H\"ormander 条件的 SDE 生成的过程的附加泛函。