论文标题
莱维过程的熵可压缩性
Entropic Compressibility of Lévy Processes
论文作者
论文摘要
与它们看似简单和共同的独立性和平稳性结构相反，莱维过程表现出各种各样的行为，从自相似的维纳过程到分段构恒定的复合泊松过程。受Ghourchian，Amini和Gohari（2018）的启发，我们通过研究了消失的离散步骤的政权来表征它们的可压缩性。对于具有绝对连续边缘的Lévy过程，这减少了在小时候了解其边缘差异熵的渐近物，为此我们获得了新的局部中央限制定理。我们将稳定过程的已知结果概括为非稳定案例，特别关注了本地自相似的莱维过程，并概念化了莱维过程的新的可压缩性层次结构，并由其blumenthal-getoor索引捕获。
In contrast to their seemingly simple and shared structure of independence and stationarity, Lévy processes exhibit a wide variety of behaviors, from the self-similar Wiener process to piecewise-constant compound Poisson processes. Inspired by the recent paper of Ghourchian, Amini, and Gohari (2018), we characterize their compressibility by studying the entropy of their double discretization (both in time and amplitude) in the regime of vanishing discretization steps. For a Lévy process with absolutely continuous marginals, this reduces to understanding the asymptotics of the differential entropy of its marginals at small times, for which we obtain a new local central limit theorem. We generalize known results for stable processes to the non-stable case, with a special focus on Lévy processes that are locally self-similar, and conceptualize a new compressibility hierarchy of Lévy processes, captured by their Blumenthal-Getoor index.
