Arijit Chakrabarty
In this paper the question of the extent to which truncated heavy tailed random vectors, taking values in a Banach space, retain the characteristic features of heavy tailed random vectors, is answered from the point of view of the central limit theorem.
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This paper investigates the decay rate of the probability that the row sum of a triangular array of truncated heavy tailed random variables is larger than an integer (k) times the truncating threshold, as both - the number of summands and the threshold go to infinity. The method of attack for this problem is significantly different from the one where k is not an integer, and requires much sharper estimates.
Fuk-Nagaev inequality in smooth Banach spaces: Optimum bounds for distributions of heavy-tailed martingales
本文在光滑Banach空间中推导了重尾martingales分布的最优Fuk-Nagaev不等式，并提出了一种新的McDiarmid型界。
First-Exit Time Analysis for Truncated Heavy-Tailed Dynamical Systems
本文分析了截断厚尾动力系统的第一退出时间，通过大偏差与元稳定性框架，揭示了相变层次和灾难原理，提出了厚尾系统的Freidlin-Wentzell理论扩展。
The empirical success of deep learning is often attributed to SGD's mysterious ability to avoid sharp local minima in the loss landscape, as sharp minima are known to lead to poor generalization. Recently, empirical evidence of heavy-tailed gradient noise was reported in many deep learning tasks, and it was shown in Şimşekli (2019a,b) that SGD can escape sharp local minima under the presence of such heavy-tailed gradient noise, providing a partial solution to the mystery. In this work, we analyze a popular variant of SGD where gradients are truncated above a fixed threshold. We show that it ac…
Well-posed Bayesian inverse problems and heavy-tailed stable quasi-Banach space priors
本文基于贝叶斯反问题框架，探讨了在准巴拿赫空间上使用厚尾稳定分布作为先验分布的可能性，并证明了在弱正则性假设下后验测度对扰动的Hellinger连续性。
Nonasymptotic Heavy-Tailed Mean Estimation in Smooth Banach Spaces
该研究拓展了Catoni和Giulini的截断均值估计器，在巴拿赫空间重尾鞅依赖、p∈(1,2]阶矩有界（含无限方差）场景下，证明了不依赖空间维度的两类时间一致界，且常数已知且小。
This paper studies the effect of truncation on the large deviations behavior of the partial sum of a triangular array coming from a truncated power law model. Each row of the triangular array consists of i.i.d. random vectors, whose distribution matches a power law on a ball of radius going to infinity, and outside that it has a light-tailed modification. The random vectors are assumed to be R^d-valued. It turns out that there are two regimes depending on the growth rate of the truncating threshold, so that in one regime, much of the heavy tailedness is retained, while in the other regime, the…
We address the important question of the extent to which random variables and vectors with truncated power tails retain the characteristic features of random variables and vectors with power tails. We define two truncation regimes, soft truncation regime and hard truncation regime, and show that, in the soft truncation regime, truncated power tails behave, in important respects, as if no truncation took place. On the other hand, in the hard truncation regime much of "heavy tailedness" is lost. We show how to estimate consistently the tail exponent when the tails are truncated, and suggest stat…
Fractional Edgeworth expansions for one-dimensional heavy-tailed random variables and applications
本文研究了一类属于 $α$-稳定随机变量吸引域的格点随机变量，它们满足截断的分数阶 Edgeworth 展开，并探讨了其运算性质、局部中心极限定理收敛速率、Green 函数展开及在离散随机偏微分方程中的应用。
Large Deviations and Metastability Analysis for Heavy-Tailed Dynamical Systems
本文为重尾随机动力系统提出大偏差与亚稳态分析框架，建立两类过程的样本路径大偏差，刻画稀有事件路径与离散层级相变，构成Freidlin-Wentzell理论的重尾对应版本。
Asymptotic theory and statistical inference for the samples problems with heavy-tailed data using the functional empirical process
本文提出截断功能经验过程（TFEP）以处理重尾数据样本问题，通过截断极端统计量稳定经验过程，推导新渐近分布并提供稳健置信区间，验证其在高斯方法失效时的有效性。
We consider a measure of dependence for symmetric $α$-stable random vectors, which was introduced by the author in 1976. We demonstrate that this measure of dependence can be extended for much more broad class of random vectors (up to regularly varying vectors in separable Banach spaces). This measure is applied for linear random processes and fields with heavy-tailed innovations, for some stable processes, and these applications show that this dependence measure, named as $α$-covariance is a good substitute for the usual covariance. Also we discuss a problem of defining long, short, and negat…
Rough Heston model as the scaling limit of bivariate cumulative heavy-tailed INAR processes: Weak-error bounds and option pricing
研究双变量累积重尾INAR(∞)过程的近不稳定特性，证明其在特定参数化和标度下收敛于粗糙Heston模型，推导出INAR不对称参数与极限波动过程杠杆相关性和扩散尺度的联系，并通过离散方程与连续分数Riccati方程比较，获得弱误差界和期权定价方法。
Constant-specified and exponential concentration inequalities play an essential role in the finite-sample theory of machine learning and high-dimensional statistics area. We obtain sharper and constants-specified concentration inequalities for the sum of independent sub-Weibull random variables, which leads to a mixture of two tails: sub-Gaussian for small deviations and sub-Weibull for large deviations from the mean. These bounds are new and improve existing bounds with sharper constants. In addition, a new sub-Weibull parameter if the italic should be retained. Please check the whole text. i…
The model of heavy Wigner matrices generalizes the classical ensemble of Wigner matrices: the sub-diagonal entries are independent, identically distributed along to and out of the diagonal, and the moments its entries are of order 1/N, where N is the size of the matrices. Adjacency matrices of Erdös-Renyi sparse graphs and matrices with properly truncated heavy tailed entries are examples of heavy Wigner matrices. We consider a family X_N of independent heavy Wigner matrices and a family Y_N of arbitrary random matrices, independent of X_N, with a technical condition (e.g. the matrices of Y_N …
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