Rafał Latała
We derive two-sided estimates on moments and tails of Gaussian chaoses, that is, random variables of the form $\sum a_{i_1,...,i_d}g_{i_1}... g_{i_d}$, where $g_i$ are i.i.d. ${\mathcal{N}}(0,1)$ r.v.'s. Estimates are exact up to constants depending on $d$ only.
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We present two-sided estimates of moments and tails of polynomial chaoses of order at most three generated by independent symmetric random variables with log-concave tails as well as for chaoses of arbitrary order generated by independent symmetric exponential variables. The estimates involve only deterministic quantities and are optimal up to constants depending only on the order of the chaos variable.
We derive two-sided estimates for random multilinear forms (random chaoses) generated by independent symmetric random variables with logarithmically concave tails. Estimates are exact up to multiplicative constants depending only on the order of chaos.
Moments and tails of Lq-valued chaoses based on independent variables with log-concave tails
本文推导了任意巴拿赫空间中二阶随机混沌矩的下界，给出了Lq空间中该混沌矩的两个上界（一个需亚高斯假设，一个无需但非最优），并可用于形状参数≥1的威布尔随机变量生成的Lq值混沌的双侧矩估计。
We derive two-sided bounds for moments and tails of random quadratic forms (random chaoses of order $2$), generated by independent symmetric random variables such that $\lVert X \rVert_{2p} \leq α\lVert X \rVert_p$ for any $p\geq 1$ and some $α\geq 1$. Estimates are deterministic and exact up to some multiplicative constants which depend only on $α$.
We show that complex hypercontractivity gives better constants than real hypercontractivity in comparison inequalities for (low) moments of Rademacher chaoses (homogeneous polynomials on the discrete cube).
We derive moment and tail estimates for Gaussian chaoses of arbitrary order with values in Banach spaces. We formulate a conjecture regarding two-sided estimates and show that it holds in a certain class of Banach spaces including L_q spaces. As a corollary we obtain two-sided bounds for moments of chaoses with values in L_q spaces based on exponential random variables.
We derive two-sided bounds for moments of random multilinear forms (random chaoses) with nonnegative coeficients generated by independent nonnegative random variables $X_i$ which satisfy the following condition on the growth of moments: $\lv X_i \rv_{2p} \leq A \lv X_i \rv_p$ for any $i$ and $p\geq 1$. Estimates are deterministic and exact up to multiplicative constants which depend only on the order of chaos and the constant $A$ in the moment assumption.
On Log-Concave-Tailed Chaoses and the Restricted Isometry Property
本文建立对数凹尾非齐次混沌过程上确界的p阶矩界及新型解耦不等式，并据此推导α-次指数元素的Hanson-Wright型偏差不等式，证明部分随机循环矩阵与时频结构随机矩阵的限制等距性质。
We prove a version of the multidimensional Fourth Moment Theorem for chaotic random vectors, in the general context of diffusion Markov generators. In addition to the usual componentwise convergence and unlike the infinite-dimensional Ornstein-Uhlenbeck generator case, another moment-type condition is required to imply joint convergence of of a given sequence of vectors.
Moment estimates for chaoses with regular moment growth
本文针对具有正则矩增长的独立对称随机变量生成的任意阶解耦混沌，给出仅依赖系数张量和一维尾部的双边矩估计，证明基于划分范数、高斯平滑与乘积比较，并补充尾部界及显式公式。
Four-moment criteria for Poisson convergence on Poisson and Rademacher chaoses
本文针对泊松与拉德马赫混沌建立泊松收敛的四矩准则，结合Chen--Stein方法等证明相关极限定理，揭示单位跳变刚性现象，明确最大影响条件的必要性，构造了对应反例与序列。
Concentration of bounded sparse chaoses and sparse Khatri-Rao embeddings
本文为稀疏有界随机变量生成的解耦齐次混沌建立矩与混合尾不等式，并应用于稀疏Khatri-Rao嵌入，给出考虑稀疏性与输入几何的保证。
In this paper, we revisit the proof of the large deviations principle of Wiener chaoses partially given by Borel, and then by Ledoux in its full form. We show that some heavy-tail phenomena observed in large deviations can be explained by the same mechanism as for the Wiener chaoses, meaning that the deviations are created, in a sense, by translations. More precisely, we prove a general large deviations principle for a certain class of functionals $f_n : \mathbb{R}^n \to \mathcal{X}$, where $\mathcal{X}$ is some metric space, under the $n$-fold probability measure $ν_α^n$, where $ν_α =Y_α^{-1}…
On the Injective Norm of Sums of Random Tensors and the Moments of Gaussian Chaoses
本文利用 PAC-Bayesian 引理给出次高斯随机张量之和期望 $\ell_p$ 单射范数的改进上界，并在 $p=2$ 时初等重现 Latała 关于高斯混沌矩的关键结果。
This paper deals with Latała's estimation of the moments of Gaussian chaoses. It is shown that his argument can be simplified significantly using Talagrand's generic chaining.
Superconvergence phenomenon in Wiener chaoses
该研究发现维纳混沌上混沌随机变量序列的正态依分布收敛可升级为超收敛，通过Malliavin分析新范式证明密度正则性提升与导数一致收敛，拓展了多元情形并给出多领域应用，去除了此前相关研究的限制性假设。
Gamma approximation and Poisson--Gaussian invariance principle on Poisson chaos
该研究在固定泊松混沌上推广伽马近似的相关估计，建立同核泊松与高斯多重积分的不变性原理，为泊松混沌的正态四矩定理提供新证明，并揭示Lindeberg条件的关键作用。
We present explicit estimates of right and left tails and exact (up to universal, multiplicative constants) estimates of tails and moments of hitting times of Bessel processes. The latter estimates are obtained from more general estimates of moments and tails established for convolutions of elementary mixtures of exponential distributions.
In this paper we obtain the non-asymptotic exact moment and tails estimates for polynomial on martingale differences. We give also some examples on order to show the exactness of obtained results.
We study how well moments of sums of independent symmetric random variables with logarithmically concave tails may be approximated by moments of Gaussian random variables.
We establish the one-to one bilateral interrelations between an asymptotic behavior for the tail of distributions for random variables and its great moments evaluation. Our results generalize the famous Richter's ones.
We derive the tail inequalities between two random variables starting from inequalities between its moment, or more generally between its Lebesgue-Riesz norms, which holds true on certain sets of parameters. We consider some applications into the martingale theory.
In this paper non-asymptotic exponential and moment estimates are derived for tail of distribution for discrete time martingale and martingale transform by means of martingale differences in the terms of moments and tails of distributions of summands and multipliers. We show also the exactness of obtained estimations and consider some applications in the theory of limit theorem for Banach space valued martingales.
Some notes on moment inequalities for heavy-tailed distributions
本文研究重尾（尤其 Pareto 型）分布的矩与尾部关系，在正则性条件下验证结论锐性，并给出任意阶 $d$ 多项式混沌的集中界。
We study the random variables (r.v.) with values in the so-called mixed (anisotropic) Lebesgue-Riesz spaces: formulate the sufficient conditions for belonging of the r.v. to these spaces, estimate the tail of norms distribution, especially deduce the exponential decreasing tails of them, etc. We obtain as a consequence the estimations of the norms of random integral operators acting between these spaces.
Asymptotics of higher-order conditional tail moments for convolution-equivalently distributed losses
本文研究系统性崩溃下高阶条件尾部矩的渐近性，针对两个投资组合的相依损失，分别处理轻尾卷积等价分布和重尾正则变化分布，证明渐近行为对相依强度稳健，并通过数值模拟验证。
We deduce the non-asymptotical (bilateral) estimates for moment inequalities for multiple sums of non-negative (more precisely, non-negative) independent random variables, on the other words, the well known U or V-statistics. Our consideration based on the correspondent estimates for the one-dimensional case by means of the so-called degenerate approximation. We apply also the theory of Bell functions as well as the properties of the Poisson distribution and the theory of the so-called Grand Lebesgue Spaces (GLS).
New Sufficient Conditions for Moment-determinacy via Probability Density Tails
本文提出一个仅基于密度函数尾部行为且易于验证的充分条件(D)，用于判定概率分布的矩确定性，并在Stieltjes和Hamburger两种情况下建立了相关定理与推论。
We derive in this preprint the moment and exponential tail estimates, sufficient conditions for the Non-Central Limit Theorem (NCLT) in the ordinary one-dimensional space as well as in the space of continuous functions for the properly (natural) normalized multi-indexed sums of function of random variables, processes or fields (r.f.), on the other words V-statistics, parametric, in general case. We construct also some examples in order to show the exactness of obtained estimates. We will use the theory of the so-called degenerate approximation of the functions of several variables as well as t…
In this paper non-asymptotic exponential and moment estimates are derived for tail of distribution for discrete time martingale under norming sequence 1/n, as in the classical Law of Large Numbers (LLN), by means of martingale differences as a rule in the terms of unconditional moments and tails of distributions of summands. We show also the exactness of obtained estimations.
In this paper non-asymptotic moment estimates are derived for tail of distribution for discrete time polynomial martingale by means of martingale differences as a rule in the terms of unconditional and unconditional relative moments and tails of distributions of summands. We show also the exactness of obtained estimations.
In a recent article the authors obtained a formula which relates explicitly the tail of risk neutral returns with the wing behavior of the Black Scholes implied volatility smile. In situations where precise tail asymptotics are unknown but a moment generating function is available we first establish, under easy-to-check conditions, tail asymptoics on logarithmic scale as soft applications of standard Tauberian theorems. Such asymptotics are enough to make the tail-wing formula work and we so obtain a version of Lee's moment formula with the novel guarantee that there is indeed a limiting slope…
We extend the class of tempered stable distributions first introduced in Rosinski 2007. Our new class allows for more structure and more variety of tail behaviors. We discuss various subclasses and the relation between them. To characterize the possible tails we give detailed results about finiteness of various moments. We also give necessary and sufficient conditions for the tails to be regularly varying. This last part allows us to characterize the domain of attraction to which a particular tempered stable distribution belongs.
A network belongs to the monotone separable class if its state variables are homogeneous and monotone functions of the epochs of the arrival process. This framework, which was first introduced to derive the stability region for stochastic networks with stationary and ergodic driving sequences, is revisited. It contains several classical queueing network models, including generalized Jackson networks, max-plus networks, polling systems, multiserver queues, and various classes of stochastic Petri nets. Our purpose is the analysis of the tails of the stationary state variables in the particular c…
This article shows a strong averaging principle for diffusions driven by discontinuous heavy-tailed Lévy noise, which are invariant on the compact horizontal leaves of a foliated manifold subject to small transversal random perturbations. We extend a result for such diffusions with exponential moments and bounded, deterministic perturbations to diffusions with polynomial moments of order $p\geq 2$, perturbed by deterministic and stochastic integrals with unbounded coefficients and polynomial moments. The main argument relies on a result of the dynamical system for each individual jump incremen…
The task for a general and useful classification of the tail behaviors of probability distributions still has no satisfactory solution. Due to lack of information outside the range of the data the tails of the distribution should be described via many characteristics. Index of regular variation is a good characteristic, but it puts too many distributions with very different tail behavior in one and the same class. One can consider for example Pareto($α$), Fr$\acute{e}$chet($α$) and Hill-horror($α$) with one and the same fixed parameter $α> 0$. The main disadvantage of VaR, expectiles, and haza…
In this paper, a critical Galton-Watson branching process with immigration $Z_{n}$ is studied. We first obtain the convergence rate of the harmonic moment of $Z_{n}$. Then the large deviation of $S_{Z_n}:=\sum_{i=1}^{Z_n} X_i$ is obtained, where $\{X_i\}$ is a sequence of independent and identically distributed zero-mean random variables with tail index $α>2$. We shall see that the converging rate is determined by the immigration mean, the variance of reproducing and the tail index of $X_1^+$, comparing to previous result for supercritical case, where the rate depends on the Schröder constant …
Absolute moments of the binomial distribution folded at its mean
该研究推导二项分布折叠后奇数阶绝对矩的精确约简式，得到有界格位移处的中心-尾展开式，将其应用于贝塔分布中位数的二阶展开，退化后可对应伽马分布中位数的相关展开项。
Asymptotics of randomly weighted sums without moment conditions of random weights
研究在无矩条件下随机加权和的渐进行为，应用于离散时间风险模型中有限时间破产概率的估计，特别处理具有常规变化尾部的增量情况。
Deviation of top eigenvalue for some tridiagonal matrices under various moment assumptions
本文研究带衰减随机势的对称三对角矩阵最大特征值的大偏差行为，在轻尾和重尾条件下推导出右尾精确估计与左尾粗略估计，并证明特定衰减速率下最大特征值服从Frechet律与确定性函数的复合分布。
Disordered systems and (subcritical) polynomial chaos with heavy-tail disorder
本文研究了受重尾无序扰动的离散统计力学系统，通过亚临界条件证明其具有非平凡标度极限，并探讨了无序相关性与系统参数的关系。
We provide a generalisation of Pinelis' Rademacher-Gaussian tail comparison to complex coefficients. We also establish uniform bounds on the probability that the magnitude of weighted sums of independent random vectors uniform on Euclidean spheres with matrix coefficients exceeds its second moment.
Exponential Tail Estimates for Multitype Poisson Branching Processes and Application to Hawkes Processes
针对多类型泊松分支过程的总后代线性泛函，建立了显式指数矩界，并应用于多类型霍克斯过程，得到高维情形下的改进结果。
Consider the Hopf--Cole solution $ h(t,x) $ of the KPZ equation with narrow wedge initial condition. Regarding $ t\to\infty $ as a scaling parameter, we provide the first rigorous proof of the Large Deviation Principle (LDP) for the lower tail of $ h(2t,0)+\frac{t}{12} $, with speed $ t^2 $ and an explicit rate function $ Φ_-(z) $. This result confirms existing physic predictions [Sasorov, Meerson, Prolhac 17], [Corwin, Ghosal, Krajenbrink, Le Doussal, Tsai 18], and [Krajenbrink, Le Doussal, Prolhac 18]. Our analysis utilizes the formula from [Borodin, Gorin 16] to convert LDP of the KPZ equat…
A Weighted Kolmogorov Metric under Sub-Cubic Moments: Admissible Parameters and an Obstruction
本文提出了一种加权 Kolmogorov 度量，用于解决高频率金融数据中厚尾特性导致的收敛问题，通过引入耗尽函数和截断方案，恢复了最优的高斯收敛速率。
In this paper non-asymptotic exponential estimates are derived for the tail distribution of polynomial martingale differences in terms unconditional tails distributions of summands. Applications are considered in the theory of polynomials on independent random variables, to the theory of U-statistics, multiply martingale series and in the theory of weak compactness measures on the Banach spaces. Partially supported by the Israel Ministry of Absorbtion Mathematics Subject Classification (2000): 47A45, 47B10, 60F10, 60G42.
We consider positively supported Borel measures for which all moments exist. On the set of compactly supported measures in this class a partial order is defined via eventual dominance of the moment sequences. Special classes are identified on which the order is total, but it is shown that already for the set of distributions with compactly supported smooth densities the order is not total. In particular we construct a pair of measures with smooth density for which infinitely many moments agree and another one for which the moments alternate infinitely often. This disproves some recently publis…
Suppose that $X$ is a bounded-degree polynomial with nonnegative coefficients on the $p$-biased discrete hypercube. Our main result gives sharp estimates on the logarithmic upper tail probability of $X$ whenever an associated extremal problem satisfies a certain entropic stability property. We apply this result to solve two long-standing open problems in probabilistic combinatorics: the upper tail problem for the number of arithmetic progressions of a fixed length in the $p$-random subset of the integers and the upper tail problem for the number of cliques of a fixed size in the random graph $…
We present order of magnitude estimates for the quantiles of non-negative linear combinations of non-negative random variables, as well as deviation inequalities for general linear combinations of independent random variables, under the assumption that all random variables satisfy the same power-type tail bound on $\mathbb{P}\{\left\vert X_i\right\vert>t\}$ of the form $t^{-q}$, $t^{-q/2}$ or $t^{-q/2}(\ln t)^{q/2}$, for $q>2$. The third type is applicable in the nonlinear setting. In the situations we consider, these results improve on classical estimates of Nagaev.
Let G_p denote the tail function of Student's distribution with p degrees of freedom. It is shown that the ratio G_q(x)/G_p(x) is decreasing in x>0 for any p and q such that 0<p<q\le\infty. Therefore, G_q(x)<G_p(x) for all such p and q and all x>0. Corollaries on the monotonicity of (generalized) moments and ratios thereof are also given.
We provide sharp bounds for the exponential moments and $p$-moments, $1\leqslant p \leqslant 2$, of the terminate distribution of a martingale whose square function is uniformly bounded by one. We introduce a Bellman function for the corresponding extremal problem and reduce it to the already known Bellman function on $\mathrm{BMO}([0,1])$. In the case of tail estimates, a similar reduction does not work exactly, so we come up with a fine supersolution that leads to sharp tail estimates.
Tail representations for absolute, signed-power, inverse, and logarithmic moments
本文建立任意随机变量实阶矩的统一框架，通过累积分布函数和生存函数给出积分表示，涵盖连续、离散和混合分布，并应用于zeta和Skellam分布，同时得到对数矩表示，连接对数均值、拉普拉斯变换与Frullani恒等式。
General upper tail estimates are given for counting edges in a random induced subhypergraph of a fixed hypergraph H, with an easy proof by estimating the moments. As an application we consider the numbers of arithmetic progressions and Schur triples in random subsets of integers. In the second part of the paper we return to the subgraph counts in random graphs and provide upper tail estimates in the rooted case.
We provide a very simple argument showing that the $Φ^4_3$ measure does have quartic exponential tails, as expected from its formal expression. This shows that the corresponding moment problem is well-posed and provides a simple path to observing non-Gaussianity of the measure.
Covariance Estimation: Optimal Dimension-free Guarantees for Adversarial Corruption and Heavy Tails
该研究提出一种协方差矩阵估计器，在对抗性样本污染与重尾分布下实现算子范数最优收敛速率，同时证明了p>4情形下的无维度Bai-Yin型定理。
Tail Profile of Bulk Gaussian Multiplicative Chaos Measures I: Bulk/Boundary Quotients
本文作为系列论文首篇，研究边界均匀奇异的体高斯乘法混沌测度右尾轮廓，建立边界Liouville共形场论中该测度体/边界商的联合矩界，提出边界局部化技巧并推广Kahane凸性不等式，为后续研究奠定基础。
Originally introduced in solid state physics to model amorphous materials and alloys exhibiting disorder induced metal-insulator transitions, the Anderson model $H_ω= -Δ+ V_ω $ on $l^2(\bZ^d)$ has become in mathematical physics as well as in probability theory a paradigmatic example for the relevance of disorder effects. Here $Δ$ is the discrete Laplacian and $V_ω = \{V_ω(x): x \in \bZ^d\}$ is an i.i.d. random field taking values in $\bR$. A popular model in probability theory is the parabolic Anderson model (PAM), i.e. the discrete diffusion equation $\partial_t u(x,t) =-H_ω u(x,t)$ on $ \bZ^…
Variational Tail Bounds for Norms of Random Vectors and Matrices
提出基于一维边际矩条件的随机向量和矩阵范数的变分尾界，通过高斯分布的推送实现聚合分布参数化，导出KL与Rényi散度组合的PAC-Bayes型界。
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