Nonasymptotic Gaussian Approximation for Inference with Stable Noise
Marina Riabiz, Tohid Ardeshiri, Ioannis Kontoyiannis, Simon Godsill
备注：
V1: 41 pages, 16 figures. V2: Text typos fixed; redundant figures from main text and appendices removed; added references in section I; changed section VI and its proofs in Appendices C-D-E; improved section IX; removed section X (Discussion and Conclusion), reference style changed. V3: title in the metadata updated. V4: Updated Theorem 1 and its proof, global revision
The results of a series of theoretical studies are reported, examining the convergence rate for different approximate representations of $α$-stable distributions. Although they play a key role in modelling random processes with jumps and discontinuities, the use of $α$-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 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, Markov chain Monte Carlo, and Particle Filtering, can then be 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 $α$-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.
Nonasymptotic and distribution-uniform Komlós-Major-Tusnády approximation
Department of Statistics and The Miller Institute, University of California, Berkeley（加州大学伯克利分校统计系及米勒研究所）；Department of Mathematical Sciences, Carnegie Mellon University（卡内基梅隆大学数学科学系）；Department of Statistics, Stanford University（斯坦福大学统计系）
A vector-contraction inequality for Rademacher complexities using $p$-stable variables
Kent State University（肯特州立大学）
Andreas Maurer在论文“A vector-contraction inequality for Rademacher complexities”中将Rademacher average的压缩不等式推广到向量值域的Lipschitz函数;他用任意idd对称和次高斯变量替换了边界表达式中的Rademacher变量。当我们用$p$-$1<p<2$的稳定变量替换亚高斯变量时，我们将看到如何扩展这项工作。
