The results of a series of theoretical studies are reported, examining the convergence rate for different approximate representations of $\alpha $ -stable distributions. Although they play a key role in modelling random processes with jumps and discontinuities, the use of $\alpha $ -stable distributions in inference often leads to analytically intractable problems. The LePage series, which is a probabilistic representation employed in this work, is used to transform an intractable, infinite-dimensional inference problem into a finite-dimensional (conditionally Gaussian) parametric problem. A major component of our approach is the approximation of the tail of this series by a Gaussian random variable. Standard statistical techniques, such as Expectation-Maximization (EM), Markov chain Monte Carlo, and Particle Filtering, can then be readily applied. In addition to the asymptotic normality of the tail of this series, we establish explicit, nonasymptotic bounds on the approximation error. Their proofs follow classical Fourier-analytic arguments, using Esséen’s smoothing lemma. Specifically, we consider the distance between the distributions of: $(i)$ the tail of the series and an appropriate Gaussian; $(ii)$ the full series and the truncated series; and $(iii)$ the full series and the truncated series with an added Gaussian term. In all three cases, sharp bounds are established, and the theoretical results are compared with the actual distances (computed numerically) in specific examples of symmetric $\alpha $ -stable distributions. This analysis facilitates the selection of appropriate truncations in practice and offers theoretical guarantees for the accuracy of resulting estimates. One of the main conclusions obtained is that, for the purposes of inference, the use of a truncated series together with an approximately Gaussian error term has superior statistical properties and is likely a preferable choice in practice.

中文翻译：

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用于具有稳定噪声的推理的非渐近高斯近似

报告了一系列理论研究的结果，检查了不同近似表示的收敛速度 $\alpha $ - 稳定的分布。尽管它们在具有跳跃和不连续性的随机过程建模中发挥着关键作用，但使用 $\alpha $ - 推理中的稳定分布通常会导致分析上难以处理的问题。LePage 级数是这项工作中使用的概率表示，用于将棘手的无限维推理问题转换为有限维（条件高斯）参数问题。我们方法的一个主要组成部分是用高斯随机变量逼近这个系列的尾部。然后可以轻松应用标准统计技术，例如期望最大化 (EM)、马尔可夫链蒙特卡罗和粒子过滤。除了该系列尾部的渐近正态性之外，我们还建立了近似误差的明确的非渐近界限。他们的证明遵循经典的傅立叶分析论证，使用 Esséen 的平滑引理。具体来说， $(i)$ 序列的尾部和适当的高斯； $(ii)$ 完整系列和截断系列；和 $(iii)$ 完整系列和截断系列加上高斯项。在所有三种情况下，都建立了明确的界限，并将理论结果与对称的具体例子中的实际距离（数值计算）进行了比较。 $\alpha $ - 稳定的分布。这种分析有助于在实践中选择适当的截断，并为结果估计的准确性提供理论保证。获得的主要结论之一是，出于推理的目的，将截断级数与近似高斯误差项一起使用具有优越的统计特性，并且在实践中可能是更可取的选择。