B. Naimark, E. Ostrovsky
We study the exact constants in the moment inequalities for sums of centered independent random variables: improve their asymptotics, low and upper bounds, calculate more exact asymptotics, elaborate the numerical algorithm for their calculation, study the class of smoothing etc.
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The aim of this paper is to propose new Rosenthal-type inequalities for moments of order higher than 2 of the maximum of partial sums of stationary sequences including martingales and their generalizations. As in the recent results by Peligrad et al. [Proc. Amer. Math. Soc. 135 (2007) 541-550] and Rio [J. Theoret. Probab. 22 (2009) 146-163], the estimates of the moments are expressed in terms of the norms of projections of partial sums. The proofs of the results are essentially based on a new maximal inequality generalizing the Doob maximal inequality for martingales and dyadic induction. Vari…
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.
Classical Kolmogorov's and Rosenthal's inequalities for the maximum partial sums of random variables are basic tools for studying the strong laws of large numbers. In this paper, motived by the notion of independent and identically distributed random variables under the sub-linear expectation initiated by Peng (2006, 2008b), we introduce the concept of negative dependence of random variables and establish Kolmogorov's and Rosenthal's inequalities for the maximum partial sums of negatively dependent random variables under the sub-linear expectations. As an application, we show that Kolmogorov's…
Sharp Rosenthal-type inequalities for mixtures and log-concave variables
本文为固定分布混合的独立随机变量和矩建立了带尖锐常数的 Rosenthal 型不等式，并在对数凹情形下确定了极值元。
We derive in this short report the exponential as well as power decreasing tail estimations for the sums of centered exchangeable random variables, alike ones for the sums of the centered independent ones.
Rosenthal-type inequalities for linear statistics of Markov chains
本文为几何遍历马尔可夫链的可加泛函建立了类似Rosenthal不等式的集中不等式，显式界依赖于混合时间，并采用基于泊松分解的新证明方法。
We deduce in this short report the non-asymptotic for exponential tail of distribution for sums of independent centered random variables.
We derive in this article the exact non-asymptotical exponential and power estimates for self-normalized sums of centered independent random variables (r.v.) under natural norming. We will use also the theory of the so-called Grand Lebesgue Spaces (GLS) of random variables.
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…
Stein's method is used to approximate sums of discrete and locally dependent random variables by a centered and symmetric Binomial distribution. Under appropriate smoothness properties of the summands, the same order of accuracy as in the Berry-Essen Theorem is achieved. The approximation of the total number of points of a point processes is also considered. The results are applied to the exceedances of the $r$-scans process and to the Matérn hardcore point process type I.
A Bernstein-type exponential inequality for (generalized) canonical U-statistics of order 2 is obtained and the Rosenthal and Hoffmann-Jørgensen inequalities for sums of independent random variables are extended to (generalized) U-statistics of any order whose kernels are either nonnegative or canonical
Let $X_1,X_2,\ldots,X_n$ be independent random variables and $S_k=\sum_{i=1}^k X_i$. We show that for any constants $a_k$, \[ \Pr(\max_{1\leq k\leq n}||S_{k}|-a_{k}|>11t)\leq 30 \max_{1\leq k\leq n}\Pr(||S_{k}|-a_{k}|>t). \] We also discuss similar inequalities for sums of Hilbert and Banach space valued random vectors.
We show, how the classical Berry-Esseen theorem for normal approximation may be used to derive rates of convergence for random sums of centerd, real-valued random variables with respect to a certain class of probability metrics, including the Kolmogorov and the Wasserstein distances. This technique is applied to several examples, including the approximation by a Laplace distribution of a geometric sum of centered random variables with finite third moment, where a concrete Berry-Esseen bound is derived. This bound reduces to a bound of the supposedly optimal order $\sqrt{p}$ in the i.i.d. case.
We prove abstract bounds on the Wasserstein and Kolmogorov distances between non-randomly centered random sums of real i.i.d. random variables with a finite third moment and the standard normal distribution. Except for the case of mean zero summands, these bounds involve a coupling of the summation index with its size biased distribution as was previously considered in \cite{GolRin96} for the normal approximation of nonnegative random variables. When being specialized to concrete distributions of the summation index like the Binomial, Poisson and Hypergeometric distribution, our bounds turn ou…
Asymptotic results for compound sums in separable Banach spaces
本文在可分离Banach空间中研究复合和序列的大偏差与中偏差，通过证明指数紧性建立结果，并分别给出被加项中心化与复合和中心化两种中偏差结论。
We consider weighted sums of independent random variables regulated by an increment sequence. We provide operative conditions that ensure strong law of large numbers for such sums to hold in both the centered and non-centered case. The existing criteria for the strong law are either implicit or assume some sufficient decay for the sequence of coefficients. In our set up we allow for arbitrary sequence of coefficients, possibly random, provided the random variables regulated by such increments satisfy some mild concentration conditions. In the non-centered case, convergence can be translated in…
We derive the sharp non-asymptotical uniform estimations for tails of distributions for classical normed sums of centered normed independent random vectors having a moderate decreasing individual tails of summands.
Concentration and moment inequalities for sums of independent heavy-tailed random matrices
本文针对仅存在低阶有限矩的独立重尾随机矩阵和，证明了依赖有效秩等内在维度的Fuk-Nagaev型与Rosenthal型不等式，并通过多个应用验证了结果优势。
We calculate the exact subgaussian norm of a centered (shifted) indicator (Bernoulli's) random variable. Using this result we derive very simple tail estimates for sums of these variables, not necessary to be identical distributed, and give some examples to show the exactness of our estimates.
We modify the classical Bernstein's inequality for the sums of independent centered random variables (r.v.) in the terms of relative tails or moments. We built also some examples in order to show the exactness of offered results.
Asymptotic results for sums and extremes
本文研究了非中心中偏差问题，证明了有上界独立同分布随机变量的和与最大值二元序列、无上界随机变量以及指数随机变量部分最小值之和的中偏差结果。
We study the Wasserstein distance of order 1 between the empirical distribution and the marginal distribution of stationary $α$-dependent sequences. We prove some moments inequalities of order p for any p $\ge$ 1, and we give some conditions under which the central limit theorem holds. We apply our results to unbounded functions of expanding maps of the interval with a neutral fixed point at zero. The moment inequalities for the Wasserstein distance are similar to the well known von Bahr-Esseen or Rosenthal bounds for partial sums, and seem to be new even in the case of independent and identic…
Exact exponential tail estimation for sums of independent centered random variables, under natural norming, with applications to the theory of U-statistics
本文利用 GLS 理论推导自然规范化独立中心化随机变量之和的精确指数尾部，并给出 U 统计量中的精细尾部估计。
We consider the self-normalized sums $T_{n}=\sum_{i=1}^{n}X_{i}Y_{i}/\sum_{i=1}^{n}Y_{i}$, where ${Y_{i} : i\geq 1}$ are non-negative i.i.d. random variables, and ${X_{i} : i\geq 1} $ are i.i.d. random variables, independent of ${Y_{i} : i \geq 1}$. The main result of the paper is that each subsequential limit law of T_n$ is continuous for any non-degenerate $X_1$ with finite expectation, if and only if $Y_1$ is in the centered Feller class.
This note investigates invariance principles for sums of N(nt) iid radom variables, where n is an integer, t is a positive real number and N(u) is a stochastic process with nonnegative integer values. We show that the sequence of sums of these random variables denoted S(n,t), when appropriately centered and normalized, weakly converges to a Gaussian process. We give sufficient conditions depending on the expectation of N(nt) which allows to rescale S(n,t) into a stochastic S(n,a(t)) weakly converging to a Brownian motion.
Normal approximation for partial sums: general convex costs
本文研究了部分和分布与正态分布之间的凸运输成本，通过理想距离和Lindeberg方法的结合，给出了非渐近界和渐近极限，并明确数值和渐近常数。
This paper derives exponential concentration inequalities and polynomial moment inequalities for the spectral norm of a random matrix. The analysis requires a matrix extension of the scalar concentration theory developed by Sourav Chatterjee using Stein's method of exchangeable pairs. When applied to a sum of independent random matrices, this approach yields matrix generalizations of the classical inequalities due to Hoeffding, Bernstein, Khintchine and Rosenthal. The same technique delivers bounds for sums of dependent random matrices and more general matrix-valued functions of dependent rand…
We compute explicit bounds in the normal and chi-square approximations of multilinear homogenous sums (of arbitrary order) of general centered independent random variables with unit variance. In particular, we show that chaotic random variables enjoy the following form of universality: (a) the normal and chi-square approximations of any homogenous sum can be completely characterized and assessed by first switching to its Wiener chaos counterpart, and (b) the simple upper bounds and convergence criteria available on the Wiener chaos extend almost verbatim to the class of homogeneous sums.
We study large partial sums, localized with respect to the sums of variances, of a sequence of centered random variables. An application is given to the distribution of prime factors of typical integers.
We calculate the so-called Rademacher's Grand Lebesgue Space norm for a centered (shifted) indicator (Bernoulli's, binary) random variable. This norm is optimal for the centered and bounded random variables (r.v.). Using this result we derive a very simple bilateral sharp exponential tail estimates for sums of these variables, not necessary to be identical distributed, under non-standard norming, and give some examples to show the exactness of our estimates.
The long-standing Gaussian product inequality (GPI) conjecture states that $E [\prod_{j=1}^{n}X_j^{2m_j}]\geq\prod_{j=1}^{n}E[X_j^{2m_j}]$ for any centered Gaussian random vector $(X_1,\dots,X_n)$ and $m_1,\dots,m_n\in\mathbb{N}$. In this paper, we describe a computational algorithm involving sums-of-squares representations of multivariate polynomials that can be used to resolve the GPI conjecture. To exhibit the power of this novel method, we apply it to prove two new GPIs: $E[X_1^{2m_1}X_2^{6}X_3^{4}]\ge E[X_1^{2m_1}]E[X_2^{6}]E[X_3^{4}]$ and $E[X_1^{2m_1}X_2^{2}X_3^{2}X_4^{2}]\ge E[X_1^{2m_…
In this paper, we establish an extension of a noncommutative Bennett inequality with a parameter $1\leq r\leq2$ and use it together with some noncommutative techniques to establish a Rosenthal inequality. We also present a noncommutative Hoeffding inequality as follows: Let $(\mathfrak{M}, τ)$ be a noncommutative probability space, $\mathfrak{N}$ be a von Neumann subalgebra of $\mathfrak{M}$ with the corresponding conditional expectation $\mathcal{E}_{\mathfrak{N}}$ and let subalgebras $\mathfrak{N}\subseteq\mathfrak{A}_j\subseteq\mathfrak{M}\,\,(j=1, \cdots, n)$ be successively independent ov…
Let $T$ be a tree with induced partial order $\preceq$. We investigate centered Gaussian processes $X=(X_t)_{t\in T}$ represented as $$ X_t=σ(t)\sum_{v \preceq t}α(v)ξ_v $$ for given weight functions $α$ and $σ$ on $T$ and with $(ξ_v)_{v\in T}$ i.i.d. standard normal. In a first part we treat general trees and weights and derive necessary and sufficient conditions for the a.s. boundedness of $X$ in terms of compactness properties of $(T,d)$. Here $d$ is a special metric defined via $α$ and $σ$, which, in general, is not comparable with the Dudley metric generated by $X$. In a second part we in…
The Laplace transform of partial sums of the square of a non-centered Gauss-Markov process, conditioning on its starting point, is explicitly computed. The parameters of multiplicative ergodicity are deduced.
We study the asymptotic behavior of the $ν$-symmetric Riemman sums for functionals of a self-similar centered Gaussian process $X$ with increment exponent $0<α<1$. We prove that, under mild assumptions on the covariance of $X$, the law of the weak $ν$-symmetric Riemman sums converge in the Skorohod topology when $α=(2\ell+1)^{-1}$, where $\ell$ denotes the smallest positive integer satisfying $\int_{0}^{1}x^{2j}ν(dx)=(2j+1)^{-1}$ for all $j=0,\dots, \ell-1$. In the case $α>(2\ell+1)^{-1}$, we prove that the convergence holds in probability.
Matrix Moment and Concentration Inequalities for Martingales and Ergodic Markov Chains with Applications in Statistical Learning
本文建立了适用于几何遍历马尔可夫链等的新型矩阵矩与集中不等式，将其推广至无限维空间并应用于马尔可夫数据的协方差估计与主成分分析，获得了更优的界。
We derive the exponential as well as power decreasing tail estimations for normed sums of centered independent identical distributed (or not) random variables on the Khintchine's form. We consider arbitrary, in particular, non-Rademacher's variables and not only Lebesgue-Riesz rearrangement invariant norms for the random variables. We intend to calculate the value of correspondent limit.
A general method for obtaining moment inequalities for functions of independent random variables is presented. It is a generalization of the entropy method which has been used to derive concentration inequalities for such functions [Boucheron, Lugosi and Massart Ann. Probab. 31 (2003) 1583-1614], and is based on a generalized tensorization inequality due to Latala and Oleszkiewicz [Lecture Notes in Math. 1745 (2000) 147-168]. The new inequalities prove to be a versatile tool in a wide range of applications. We illustrate the power of the method by showing how it can be used to effortlessly re-…
Limit theorems for squared increment sums of the maximum of two isotropic fractional Brownian fields under fixed-domain asymptotics
研究两个独立同分布各向同性分数布朗运动场在固定二维域上的点wise最大值的平方增量和，揭示其在Hurst参数H<1/2时的渐近行为，证明归一化后收敛于确定常数乘以两个场差的局部时间。
A Sharper Hoeffding Bound for Weighted Sums of Exchangeable Random Variables
本文针对有界可交换随机变量加权和，证明了更紧的Hoeffding型矩生成函数界，改进了现有膨胀因子的阶并给出显式常数，还证明了逆总体大小膨胀的不可避免性。
We study some SDEs derived from the $q\to 1$ limit of a 2D surface growth model called the $q$-Whittaker process. The fluctuations are proven to exhibit Gaussian characteristics that "come down from infinity": After rescaling and re-centering, convergence to the time-inverted stationary additive stochastic heat equation holds. The point of view in this paper is a probabilistic representation of the SDEs by independent sums. By this connection, the normal and Poisson approximations and the in-between slow decorrelation, all in particular integrated forms, explain the convergence of the re-cente…
We study the convergence of centered and normalized sums of i.i.d. random elements of the space $\mathcal{D}$ of c{á}dl{á}g functions endowed with Skorohod's $J\_1$ topology, to stable distributions in $\mathcal D$. Our results are based on the concept of regular variation on metric spaces and on point process convergence. We provide some applications, in particular to the empirical process of the renewal-reward process.
The famous results of Komlós, Major and Tusnády (see [15] and [17]) state that it is possible to approximate almost surely the partial sums of size n of i.i.d. centered random variables in L p (p > 2) by a Wiener process with an error term of order o(n 1/p). Very recently, Berkes, Liu and Wu [3] extended this famous result to partial sums associated with functions of an i.i.d. sequence, provided a condition on a functional dependence measure in L p is satisfied. In this paper, we adapt the method of Berkes, Liu and Wu to partial sums of functions of random iterates. Taking advantage of the Mar…
We study random exponential sums of the form $\sum_{k=1}^nX_k\times\ex p\{i(λ_k^{(1)}t_1+...+λ_k^{(s)}t_s)\}$, where $\{X_n\}$ is a sequence of random variables and $\{λ_n^{(i)}:1\leq i\leq s\}$ are sequences of real numbers. We obtain uniform estimates (on compact sets) of such sums, for independent centered $\{X_n\}$ or bounded $\{X_n\}$ satisfying some mixing conditions. These results generalize recent results of Weber [Math. Inequal. Appl. 3 (2000) 443--457] and Fan and Schneider [Ann. Inst. H. Poincaré Probab. Statist. 39 (2003) 193--216] in several directions. As applications we derive c…
Let $X$ be a centered random variable with unit variance, zero third moment, and such that $E[X^4] \ge 3$. Let $\{F_n : n\geq 1\}$ denote a normalized sequence of homogeneous sums of fixed degree $d\geq 2$, built from independent copies of $X$. Under these minimal conditions, we prove that $F_n$ converges in distribution to a standard Gaussian random variable if and only if the corresponding sequence of fourth moments converges to $3$. The statement is then extended (mutatis mutandis) to the free probability setting. We shall also discuss the optimality of our conditions in terms of explicit t…
Elementary Symmetric Polynomial Inequalities for Centered Vectors and Matrices
研究和为零的向量及行列和为零的方阵的初等对称多项式不等式，应用这些结果得到排列混合平均场近似保证的统一上界及德·菲内蒂定理的精确卡方版本，证明思路由GPT - 5.5 Pro模型提出。
Let $f_1, f_2, ..., f_n$ be a family of independent copies of a given random variable f in a probability space $(Ω, \mathcal{F}, μ)$. Then, the following equivalence of norms holds whenever $1 \le q \le p < \infty$ $$\Big(\int_Ω \Big[ \sum_{k=1}^n |f_k|^q \Big]^{\frac{p}{q}} d μ\Big)^{\frac1p} \sim \max_{r \in \{p,q\}} {n^{\frac1r} \Big(\int_Ω|f|^r dμ\Big)^{\frac1r}}.$$ We prove a noncommutative analogue of this inequality for sums of free random variables over a given von Neumann subalgebra. This formulation leads to new classes of noncommutative function spaces which appear in quantum probab…
Real Gaussian exponential sums via a real moment map
本文通过实矩映射研究高斯指数和系统的期望解数，将其化为牛顿多胞体上的积分，并证明添加支撑点可导致解数减少，否定局部猜想，同时给出Aronszajn乘法新下界。
Take a centered random walk S_n and consider the sequence of its partial sums A_n = S_1 + ... + S_n. Suppose S_1 is in the domain of normal attraction of an α-stable law with 1 < α<= 2. Assuming that S_1 is either right-exponential (that is P(S > x | S > 0)=e^{-ax} for some a > 0 and all x > 0) or right-continuous (skip free), we prove that p_N = P(A_1 > 0, ..., A_N > 0) ~ C_αN^{1/(2α) - 1/2} as N tends to infinity, where C_α> 0 depends on the distribution of the walk. We also consider a conditional version of this problem and study positivity of integrated discrete bridges.
In this paper we show how to apply classical probabilistic tools for partial sums $\sum_{j=0}^{n-1}φ\circτ^j$ generated by a skew product $τ$, built over a sufficiently well mixing base map and a random expanding dynamical system. Under certain regularity assumptions on the observable $φ$, we obtain a central limit theorem (CLT) with rates, a functional CLT, an almost sure invariance principle (ASIP), a moderate deviations principle, several exponential concentration inequalities and Rosenthal type moment estimates for skew products with $α, ϕ$ or $ψ$ mixing base maps and expanding on the aver…
Let $ V_{n} = X_{1,n} + X_{2,n} + \cdots + X_{n,n}$ where $X_{i,n}$ are Bernoulli random variables which take the value $1$ with probability $b(i;n)$. Let $λ_{n} = \sum\limits_{i=1}^{n} b(i;n) $, $λ= \lim\limits_{n \to \infty} λ_n,$ and $m_n = \max\limits_{1 \leq i \leq n} b(i;n)$. We derive asymptotic results for $P(V_{n}=k)$ that hold without assuming that $λ< +\infty$ or $m_n \to 0$. Also, we do not assume $k$ to be fixed, but instead, our results hold uniformly for all $k$ which satisfy particular growth conditions with respect to $n$. These results extend known Poisson local limit theorem…
We consider a finite impulse response system with centered independent sub-Gaussian design covariates and noise components that are not necessarily identically distributed. We derive non-asymptotic near-optimal estimation and prediction bounds for the least-squares estimator of the parameters. Our results are based on two concentration inequalities on the norm of sums of dependent covariate vectors and on the singular values of their covariance operator that are of independent value on their own and where the dependence arises from the time shift structure of the time series. These results gen…
Effective-dimension bounds in the central limit theorem for centered Euclidean balls
该研究针对独立随机向量和的中心化欧氏球，建立了中心极限定理的非渐近高斯近似界，结合有效秩依赖与n^{-1}速率，可推广到可分希尔伯特空间，证明基于傅里叶变换等方法。
Annealed limit for a diffusive disordered mean-field model with random jumps
研究随机环境中由点过程驱动且具有无序跳跃高度的平均场粒子系统，证明其依分布收敛于由高斯变量驱动的 SDE 极限过程，并给出有限维分布的收敛速度。
Permutations from Random Walk
本文研究随机游走部分和排序产生的排列，证明其分布与步长分布无关，并推广至一般情形，引入Wiener permuton，给出插值构造，统一扩展了反正弦定律。
Self-normalized processes arise naturally in statistical applications. Being unit free, they are not affected by scale changes. Moreover, self-normalization often eliminates or weakens moment assumptions. In this paper we present several exponential and moment inequalities, particularly those related to laws of the iterated logarithm, for self-normalized random variables including martingales. Tail probability bounds are also derived. For random variables B_t>0 and A_t, let Y_t(λ)=\exp{λA_t-λ^2B_t^2/2}. We develop inequalities for the moments of A_t/B_{t} or sup_{t\geq 0}A_t/{B_t(\log \log B_{…
Deviation bounds for the norm of a random vector under exponential moment conditions with applications
本文在指数矩条件下推广 Hanson-Wright 不等式，给出随机向量范数的上分位函数，并揭示相变效应，应用于伯努利向量和与协方差估计。
Improved Berry-Esseen bounds for multivariate nonlinear statistics in convex distance
本文针对多变量非线性统计量在凸距离下建立两个非渐近Berry-Esseen界，改进维数增长条件，并应用于随机逼近、时间差分学习和U-统计量，给出显式高斯近似误差及收敛充分条件。
Under an appropriate regular variation condition, the affinely normalized partial sums of a sequence of independent and identically distributed random variables converges weakly to a non-Gaussian stable random variable. A functional version of this is known to be true as well, the limit process being a stable Lévy process. The main result in the paper is that for a stationary, regularly varying sequence for which clusters of high-threshold excesses can be broken down into asymptotically independent blocks, the properly centered partial sum process still converges to a stable Lévy process. Due …
Breuer-Major Theorems for Hilbert Space-Valued Random Variables
针对Hilbert空间值平稳高斯过程的子过程，利用Hilbert空间上的第四矩定理和无穷维Malliavin-Stein框架，建立了中心极限定理及其连续时间和定量版本，并应用于函数数据分析和神经算子。
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