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首页> 外文期刊>Annales de l'Institut Henri Poincare. Probabilites et Statistiques >Total variation distance for discretely observed Levy processes: A Gaussian approximation of the small jumps
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Total variation distance for discretely observed Levy processes: A Gaussian approximation of the small jumps

机译：离散观察征收过程的总变化距离：小跳跃的高斯近似

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摘要

It is common practice to treat small jumps of Levy processes as Wiener noise and to approximate its marginals by a Gaussian distribution. However, results that allow to quantify the goodness of this approximation according to a given metric are rare. In this paper, we clarify what happens when the chosen metric is the total variation distance. Such a choice is motivated by its statistical interpretation; if the total variation distance between two statistical models converges to zero, then no test can be constructed to distinguish the two models and they are therefore asymptotically equally informative. We elaborate a fine analysis of a Gaussian approximation for the small jumps of Levy processes in total variation distance. Non-asymptotic bounds for the total variation distance between n discrete observations of small jumps of a Levy process and the corresponding Gaussian distribution are presented and extensively discussed. As a byproduct, new upper bounds for the total variation distance between discrete observations of Levy processes are provided. The theory is illustrated by concrete examples.
机译：通常的做法是将征收过程的小跳跃作为维纳噪声处理，并通过高斯分布近似其边缘。然而，允许量化根据给定度量的近似的良好的结果是罕见的。在本文中，我们阐明了所选度量是总变化距离时会发生什么。这种选择是通过其统计解释的动机;如果两个统计模型之间的总变化距离会聚到零，则不能构造任何测试以区分两个模型，因此它们是渐近的相同信息。我们详细说明了对总变化距离的征收过程的小跳跃的高斯近似的精细分析。呈现和广泛地讨论了N个离散观察的N离散观察的总变化的非渐近界限和相应的高斯分布。作为副产品，提供了征收过程离散观察之间的总变化距离的新上限。该理论由具体实施例说明。

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