Radosław Adamczak, Rafał Latała, Rafał Meller
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.
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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 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 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.
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 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 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.
We develop a theory of Malliavin calculus for Banach space valued random variables. Using radonifying operators instead of symmetric tensor products we extend the Wiener-Ito isometry to Banach spaces. In the white noise case we obtain two sided L^p-estimates for multiple stochastic integrals in arbitrary Banach spaces. It is shown that the Malliavin derivative is bounded on vector-valued Wiener-Ito chaoses. Our main tools are decoupling inequalities for vector-valued random variables. In the opposite direction we use Meyer's inequalities to give a new proof of a decoupling result for Gaussian …
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.
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。
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。
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。
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。
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.
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 the article, integration of temporal functions in (possibly non-UMD) Banach spaces with respect to (possibly non-Gaussian) fractional processes from a finite sum of Wiener chaoses is treated. The family of fractional processes that is considered includes, for example, fractional Brownian motions of any Hurst parameter or, more generally, fractionally filtered generalized Hermite processes. The class of Banach spaces that is considered includes a large variety of the most commonly used function spaces such as the Lebesgue spaces, Sobolev spaces, or, more generally, the Besov and Lizorkin-Tri…
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。
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.
We prove a Marcinkiewicz-Zygmund type inequality for random variables taking values in a smooth Banach space. Next, we obtain some sharp concentration inequalities for the empirical measure of $\{T, T^2, \cdots, T^n\}$, on a class of smooth functions, when $T$ belongs to a class of nonuniformly expanding maps of the unit interval.
Noncommutative Regularity Structures
本文推广正则性结构理论至局部$m$-凸拓扑代数，涵盖局部$C^*$-代数及$q$-高斯过程，并引入新范数解决非交换重整化估计挑战。
Certain previously known upper bounds on the moments of the norm of martingales in 2-smooth Banach spaces are improved. Some of these improvements hold even for sums of independent real-valued random variables. Applications to concentration of measure on product spaces for separately Lipschitz functions are presented, including ones concerning the central moments of the norm of the sums of independent random vectors in any separable Banach space.
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。
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…
A general device is proposed, which provides for extension of exponential inequalities for sums of independent real-valued random variables to those for martingales in the 2-smooth Banach spaces. This is used to obtain optimum bounds of the Rosenthal-Burkholder and Chung types on moments of the martingales in the 2-smooth Banach spaces. In turn, it leads to best-order bounds on moments of the sums of independent random vectors in any separable Banach spaces. Although the emphasis is put on the infinite-dimensional martingales, most of the results seem to be new even for the one-dimensional one…
Stochastic Volterra equations with random functional coefficients in Banach spaces
本文提出了一种新的方法，用于推导带有随机函数系数的随机Volterra方程在Banach空间中的唯一解，并证明了在特定条件下这些解的强解性质。
Stochastic Approximation in Banach Spaces Without Geometric Constraints
研究在巴拿赫空间中，噪声序列为均值0的独立同分布序列或满足适当矩条件的独立随机变量序列时的随机逼近，无需空间几何约束，证明相关结论及条件。
Let $(X_i, \mathcal{F}_i)_{i\geq1}$ be a martingale difference sequence in a smooth Banach space. Let $S_n=\sum_{i=1}^nX_i, n\geq 1,$ be the partial sums of $(X_i, \mathcal{F}_i)_{i\geq 1}$. We give upper bounds on the quantity $\mathbb{P}\left(\max_{1\leq k\leq n}\lVert S_k\rVert>nx\right)$ in terms of $ n\geq 1$ and $x>0$ in two different situations: when the martingale differences have uniformly bounded exponential moments and when the decay of the tail of the increments is polynomial.
We define the $k$:th moment of a Banach space valued random variable as the expectation of its $k$:th tensor power; thus the moment (if it exists) is an element of a tensor power of the original Banach space. We study both the projective and injective tensor products, and their relation. Moreover, in order to be general and flexible, we study three different types of expectations: Bochner integrals, Pettis integrals and Dunford integrals. One of the problems studied is whether two random variables with the same injective moments (of a given order) necessarily have the same projective moments; …
This paper studies topological duals of Banach function spaces (BFS). We assume a finite measure but our arguments extend to general locally convex function spaces whose topology is generated by seminorms that satisfy the usual BFS axioms. The dual is identified with the direct sum of another space of random variables (Köthe dual), a space of purely finitely additive measures and the annihilator of $L^\infty$. In the special case of rearrangement invariant spaces, the second component in the dual vanishes and we obtain various classical as well as new duality results e.g. on Lebesgue, Orlicz, …
Deviation and moment inequalities for Banach-valued $U$-statistics
本文在可分 Banach 空间光滑性假设下建立了独立数据 U 统计量的偏差不等式，并将其用于大数定律速率、Hölder 型泛函中心极限定理和不完全 U 统计量的矩不等式。
We consider decoupling inequalities for random variables taking values in a Banach space $X$. We restrict the class of distributions that appear as conditional distributions while decoupling and show that each adapted process can be approximated by a Haar type expansion in which only the same conditional distributions appear. Moreover, we show that in our framework a progressive enlargement of the underlying filtration does not effect the decoupling properties (e.g., the constants involved). As special case we deal with one-sided moment inequalities when decoupling dyadic (i.e., Paley-Walsh) m…
When a spatial process is recorded over time and the observation at a given time instant is viewed as a point in a function space, the result is a time series taking values in a Banach space. To study the spatio-temporal extremal dynamics of such a time series, the latter is assumed to be jointly regularly varying. This assumption is shown to be equivalent to convergence in distribution of the rescaled time series conditionally on the event that at a given moment in time it is far away from the origin. The limit is called the tail process or the spectral process depending on the way of rescali…
Let $X$ be a symmetric quasi-Banach function space with Fatou property and let $E$ be an arbitrary symmetric quasi-Banach sequence space. Suppose that $(f_k)_{k\geq0}\subset X$ is a sequence of independent random variables. We present a necessary and sufficient condition on $X$ such that the quantity $$\Big\|\ \Big\|\sum_{k=0}^nf_ke_k\Big\|_{E}\ \Big\|_X$$ admits an equivalent characterization in terms of disjoint copies of $(f_k)_{k=0}^n$ for every $n\ge 0$; in particular, we obtain the deterministic description of $$\Big\|\ \Big\|\sum _{k=0}^nf_ke_k\Big\|_{\ell_q}\ \Big\|_{L_p}$$ for all $0<…
A Banach space valued graphon is a function $W:(Ω, \mathcal{A},π)^2\to\mathcal{Z}$ from a probability space to a Banach space with a separable predual, measurable in a suitable sense, and lying in appropriate $L^p$-spaces. As such we may consider $W(x,y)$ as a two-variable random element of the Banach space. A two-dimensional analogue of moments can be defined with the help of graphs and weak-* evaluations, and a natural question that then arises is whether these generalized moments determine the function $W$ uniquely -- up to measure preserving transformations. The main motivation comes from …
We discuss two-sided bounds for moments and tails of quadratic forms in Gaussian random variables with values in Banach spaces. We state a natural conjecture and show that it holds up to additional logarithmic factors. Moreover in a certain class of Banach spaces (including $L_r$-spaces) these logarithmic factors may be eliminated. As a corollary we derive upper bounds for tails and moments of quadratic forms in subgaussian random variables, which extend the Hanson-Wright inequality.
In this paper, we will study concentration inequalities for Banach space-valued martingales. Firstly, we prove that a Banach space $X$ is linearly isomorphic to a $p$-uniformly smooth space ($1<p\leq 2$) if and only if an Azuma-type inequality holds for $X$-valued martingales. This can be viewed as a generalization of Pinelis' work on Azuma inequality for martingales with values in $2$-uniformly smooth space. Secondly, Azuma-type inequality for self-normalized sums will be presented. Finally, some further inequalities for Banach space-valued martingales, such as moment inequalities for double …
A general Kolmogorov-Chentsov type theorem on general metric spaces with applications to limit theorems for Banach-valued processes
本文推广 Kolmogorov-Chentsov 定理，在一般度量空间中利用 Talagrand 链式技术建立连续模与 Hölder 修正判据，并用于证明 Banach 值随机过程的一致胎紧性和中心极限定理。
Monte Carlo convergence rates for $k$th moments in Banach spaces
研究Banach空间中k阶矩的蒙特卡洛收敛速率，推导标准和多级方法的误差估计及计算成本优化。
Let X be a Banach space. We prove p-independence of the one-sided decoupling inequality for X-valued tangent martingales as introduced by Kwapien and Woyczynski. It is known that a Banach space X satisfies the two-sided decoupling inequality if and only if X is a UMD Banach space. The one-sided decoupling inequality is a weaker property, including e.g. the space L^1. We provide information on the optimal constants for various spaces, and give a upper estimate of order p in general. In the second part of our paper we derive Burkholder-Davis-Gundy type estimates for p-th moments, p in (0,infty),…
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.
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。
We study the Banach space $D([0,1]^m)$ of functions of several variables that are (in a certain sense) right-continuous with left limits, and extend several results previously known for the standard case $m=1$. We give, for example, a description of the dual space, and we show that a bounded multilinear form always is measurable with respect to the $σ$-field generated by the point evaluations. These results are used to study random functions in the space. (I.e., random elements of the space.) In particular, we give results on existence of moments (in different senses) of such random functions,…
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。
A Functional Central Limit Theorem for Locally Stationary Time Series in Banach Spaces
本文为Banach空间中的局部平稳时间序列建立了函数中心极限定理，不依赖type-2或cotype假设，并应用于自归一化CUSUM变点检测。
We introduce polynomial processes taking values in an arbitrary Banach space $B$ via their infinitesimal generator $L$ and the associated martingale problem. We obtain two representations of the (conditional) moments in terms of solutions of a system of ODEs on the truncated tensor algebra of dual respectively bidual spaces. We illustrate how the well-known moment formulas for finite dimensional or probability-measure valued polynomial processes can be deduced in this general framework. As an application we consider polynomial forward variance curve models which appear in particular as Markovi…
Measure-Valued CARMA Processes
本文提出由 Lévy 从属过程驱动的测度值 CARMA 过程，证明其存在性、锥不变性与平稳条件，并展示其对时空随机场泛函建模的适用性。
Well-posedness of Bayesian inverse problems in quasi-Banach spaces with stable priors
本文研究了在无限维参数空间中贝叶斯逆问题的适定性，探讨了在稳定分布先验下的逆问题求解方法及后验测度的连续性。
Small ball inequalities have been extensively studied in the setting of Gaussian processes and associated Banach or Hilbert spaces. In this paper, we focus on studying small ball probabilities for sums or differences of independent, identically distributed random elements taking values in very general sets. Depending on the setting--abelian or nonabelian groups, or vector spaces, or Banach spaces--we provide a collection of inequalities relating different small ball probabilities that are sharp in many cases of interest. We prove these distribution-free probabilistic inequalities by showing th…
Optimized multilevel Monte Carlo methods in Banach spaces
针对Banach空间值随机变量的矩估计，通过精细误差分析考虑Rademacher类型常数对维度的依赖，提出优化单级和多级蒙特卡洛方法，在L^p(S)值情形下实现与Rademacher类型无关的收敛率。
Stochastic Partial Differential Equations Associated with Pseudo-Differential Operators and Hilbert Space-Valued Gaussian Processes
本文研究了Hilbert空间中由Gaussian过程驱动、与伪微分算子相关的随机偏微分方程，证明了其解的唯一存在性并建立了$L^{p}$-正则性框架，主要贡献包括建立$p$-阶矩极大不等式与Littlewood-Paley型不等式，并改进了Fourier乘子充分条件。
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