On superconvergence of sums of free random variables
Vladislav Kargin
本文给出了有界自由随机变量和超收敛的充分条件，并给出了超收敛速度的估计。
This paper derives sufficient conditions for superconvergence of sums of bounded free random variables and provides an estimate for the rate of superconvergence.
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The phenomenon of superconvergence is proved for all freely infinitely divisible distributions. Precisely, suppose that the partial sums of a sequence of free identically distributed, infinitesimal random variables converge in distribution to a nondegenerate freely infinitely divisible law. Then the distribution of the sum becomes Lebesgue absolutely continuous with a continuous density in finite time, and this density can be approximated by that of the limit law uniformly, as well as in all $L^{p}$-norms for $p>1$, on the real line except possibly in the neighborhood of one point. Application…
The superconvergence phenomenon is shown for products of free, identically distributed random variables. We also show that a certain Holder regularity, first demonstrated by Biane for the density of a free additive convolution with a semicircular law, extends to free additive and multiplicative convolutions with arbitrary freely infinitely divisible laws and to free convolution semigroups.
This paper describes the quality of convergence to an infinitely divisible law relative to free multiplicative convolution. We show that convergence in distribution for products of identically distributed and infinitesimal free random variables implies superconvergence of their probability densities to the density of the limit law. Superconvergence to the marginal law of free multiplicative Brownian motion at a specified time is also studied. In the unitary case, the superconvergence to free Brownian motion and that to the Haar measure are shown to be uniform over the entire unit circle, imply…
We extend, in the free probability framework, an invariance principle for multilinear homogeneous sums with low influences recently established in [E. Mossel, R. O'Donnell and K. Oleszkiewicz (2010). Noise stability of functions with low influences: invariance and optimality. {\it Ann. Math.} {\bf 171}, no. 1, 295-341]. To do so, a hypercontractivity property for those homogeneous sums is necessary, and to prove it has turned out to be our main task. Finally, we deduce from our extension several universality phenomenons, in the spirit of the paper [I. Nourdin, G. Peccati and G. Reinert (2010).…
The optimal order for the p-th moment of sums of independent random variables with respect to symmetric norms and related combinatorial estimates
我们计算p-n个独立随机变量之和相对于R^n中对称范数的矩。在此估计中得到的上界p/ln p的增长顺序是最优的。在对f的适当假设下，将结果推广到广义Lorentz空间l_{f,w}.事实上，关键的组合估计得到的弱l_1（l_{1,infinity}）-范数。类似的结果已经获得独立的戈登，Litvak，Schuett和沃纳Orlicz规范和蒙哥马利史密斯使用不同的技术和避免组合估计。
We give explicit bounds for the tail probabilities for sums of independent geometric or exponential variables, possibly with different parameters.
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.
In this paper, we obtain error bound for binomial and negative binomial approximations to weighted sums of locally dependent random variables, using Stein's method. We also discuss approximation results for weighted sums of independent random variables. We demonstrate our results through some applications in finance and runs in statistics.
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 obtain decay rates of probabilities of tails of polynomials in several independent random variables with heavy tails and derive stable limit theorems for nonconventional sums of such polynomials
Known Bernstein-type upper bounds on the tail probabilities for sums of independent zero-mean sub-exponential random variables are improved in several ways at once. The new upper bounds have a certain optimality property.
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.
We deduce in this short report the non-asymptotic for exponential tail of distribution for sums of independent centered random variables.
We study the large deviation probabilities of infinite weighted sums of independent random variables that have stretched exponential tails. This generalizes Kiesel and Stadtmüller (2000), who study the same objects under the assumption of finite exponential moments, and Gantert et al.\ (2014), who study finite weighted sums with stretched exponential tails.
Exact Constants in the Rosenthal Moment Inequalities for Sums of independent centered Random Variables
本文研究了中心独立随机变量和的矩不等式中的精确常数：改进了它们的渐近性、上下界上界，计算了更精确的渐近性，详细阐述了计算它们的数值算法，研究了光滑类等。
The limit theorems on extreme order statistics and partial sums of i.i.d. random variables
该研究针对独立同分布随机变量，证明了其极值顺序统计量与部分和联合形式的弱极限定理，并将结果拓展至几乎必然极限情形。
A note on the invariance principle of the product of sums of random variables
文献中研究了各种随机变量和乘积的中心极限定理。本文旨在说明这类结果是不变原理的推论。
In this paper, we prove a Berry--Esseen bound with optimal order for self-normalized sums of local dependent random variables under some mild dependence conditions. The proof is based on Stein's method and a randomized concentration inequality. As applications, we obtain optimal Berry--Esseen bounds for $m$-dependent random variables and graph dependency.
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.
We continue the research of Latała on improving estimates of $p$-th moments of sums of independent random variables. We generalize some of his results in the case when $2 \leq p \leq 4$ and present a combinatorial approach for even moments.
In this paper, we study the fluctuations of sums of random variables with distribution defined as a mixture of light-tail and truncated heavy-tail distributions. We focus on the case when both the mixing coefficient and the truncation level depend on the number of summands. The aim of this research is to characterize the limiting distributions of the sums due to various relations between these parameters.
Tail bounds for sums of independent two-sided exponential random variables
本文为双边指数随机变量加权和建立匹配主项的尾部上下界，推广了Janson的单边指数结果。
Limit Theorems for Sums of p-Adic Random Variables
本文研究了稳定分布的p-adic对应分布，即独立同分布p-adic值随机变量的规范化和序列的极限分布。与经典的情况相反，非退化极限分布只能在某些假设下获得的渐近行为的数目的被加数的近似和。这种渐进性决定了“稳定性指数”。
Poisson Approximation to Weighted Sums of Locally Dependent Random Variables
该研究利用 Stein 方法分析局部相依随机变量加权和的 Poisson 逼近，给出优于已有结论的误差界，并通过数值比较验证其有效性。
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.
Conditional large and moderate deviations for sums of discrete random variables. Combinatoric applications
我们证明了大和中等偏差原则的经验平均值的离散独立同分布的总和的值为条件的分布。随机变量。讨论了组合问题的一些应用。
This paper develops Rio's method [C. R. Acad. Sci. Paris Sér. I Math., 1995] to prove the weak law of large numbers for maximal partial sums of pairwise independent random variables. The method allows us to avoid using the Kolmogorov maximal inequality. As an application, a weak law of large numbers for maximal partial sums of pairwise independent random variables under a uniform integrability condition is also established. The sharpness of the result is illustrated by an example.
We investigate the complete $p$-th moment convergence for weighted sums of independent, identically distributed random variables under sublinear expectations space. Using moment inequality and truncation methods, we prove the equivalent conditions of complete $p$-th moment convergence of weighted sums of independent, identically distributed random variables under sublinear expectations space, which complement the corresponding results obtained in Guo and Shan (2020).
In this paper, based on the initiation of the notion of negatively associated random variables under nonlinear probability, a strong limit theorem for weighted sums of random variables within the same frame is achieved without assumptions of independence and identical distribution, from which the Marcinkiewich-Zygmund type and Kolmogorov type strong laws of large numbers are derived. In addition, as applications of our results, Stranssen type invariance principles of negatively associated random variables and vertically independent random variables are proposed respectively.
We consider complete convergence and closely related Hsu-Robbins-Erdos-Spitzer-Baum-Katz series for sums whose terms are elements of a linear 2-nd order autoregressive sequences of random variables and prove sufficient conditions for the convergence of this series.
In this paper, we obtain some results on precise large deviations for non-random and random sums of widely dependent random variables with common dominatedly varying tail distribution or consistently varying tail distribution on $(-\infty,\infty)$. Then we apply the results to reinsurance and insurance and give some asymptotic estimates on proportional reinsurance, random-time ruin probability and the finite-time ruin probability.
In this article, we consider Poisson and Poisson convoluted geometric approximation to the sums of $n$ independent random variables under moment conditions. We use Stein's method to derive the approximation results in total variation distance. The error bounds obtained are either comparable to or improvement over the existing bounds available in the literature. Also, we give an application to the waiting time distribution of 2-runs.
The complete convergence for weighted sums of sequences of independent, identically distributed random variables under sublinear expectations space was studied. By moment inequality and truncation methods, we establish the equivalent conditions of complete convergence for weighted sums of sequences of independent, identically distributed random variables under sublinear expectations space. The results extend the corresponding results obtained by Guo (2012) to those for sequences of independent, identically distributed random variables under sublinear expectations space.
Skellam compound Poisson approximation to the sums of symmetric Markov dependent random variables
研究对称马尔可夫相依三点随机变量之和的近似问题，用Skellam随机变量近似该和，并在局部、总变差和Wasserstein度量下估计精度，利用测度卷积性质完成证明。
Concentration of normalized sums and a central limit theorem for noncorrelated random variables
对于互不相关的随机变量，我们研究了由给定长的时间序列构成的归一化和分布族的集中性质。
Convergence for sums of i. i. d. random variables under sublinear expectations
本文研究了次线性期望空间下独立同分布随机变量部分加权和的收敛性问题，证明了最大值完全矩收敛的等价条件，并建立了Baum-Katz型结果，扩展了经典线性期望下的相关理论。
We obtain some optimal inequalities on tail probabilities for sums of independent bounded random variables. Our main result completes an upper bound on tail probabilities due to Talagrand by giving a one-term asymptotic expansion for large deviations. This result can also be regarded as sharp large deviations of types of Cramér and Bahadur-Ranga Rao.
Moderate Deviations of Hitting Times for Trajectories of Sums of Independent and Identically Distributed Random Variables
本文研究独立同分布随机变量和的轨迹首达时的中偏差问题，通过区间转单点的思路结合已有大、中偏差结果，建立了对应的中偏差原理并证明转化不改变速率函数。
In this work we prove an asymptotic result, that under some conditions on the involved distribution functions, is valid for any Oppenheim expansion, extending a classical result proven by W. Vervaat in 1972 for denominators of the Luroth case. Furthermore, we study the convergence in distribution of weighted sums of a sequence of independent random variables. Although the result is of its own interest, in the present setting it is used to prove convergence in distribution of specific sequences of random variables generalizing known results obtained for Luroth random variables.
Lower bounds for tails of sums of independent symmetric random variables
本文推广了Kleitman（1970）和Kanter（1976）对多元集中函数不等式的处理方法，得到了独立对称随机变量和的偏差概率的一个仅依赖于和项偏差概率的下界。这个界在离散效应下是最优的，改进了Nagaev（2001）的结果，并补充了Birnbaum（1948）和Pruss（1997）的比较定理。将单峰随机变量的Birnbaum定理推广到格型情形。
On Sums of Conditionally Independent Subexponential Random Variables
独立次指数随机变量和的渐近尾行为是很好理解的，主要特征之一是单大跳跃原理。我们研究的情况下，依赖次指数随机变量，确定性和随机总和，使用一种新的方法，通过考虑条件独立结构的随机变量。我们寻求独立随机变量理论的结果仍然成立的充分条件。对于次指数分布，我们引入了边界函数类的概念，我们希望这将是一个有用的工具，在研究次指数随机变量的许多方面。我们在论文中给出的例子证明了由于依赖性的各种影响，并且它们本身也很有趣。
We obtain the analogue of the classical result by Erdös and Kac on the limiting distribution of the maximum of partial sums for exchangeable random variables with zero mean and variance one. We show that, if the conditions of the central limit theorem of Blum et al. hold, the limit coincides with the classical one. Under more general assumptions, the probability of the random variables having conditional negative drift appears in the limiting distribution.
Central limit theorem for sampled sums of dependent random variables
在弱相关条件下证明了线性三角阵的一个中心极限定理。我们的结果，然后应用到研究的依赖随机变量采样的$\bbZ$值的瞬态随机游动。这扩展了Guillotin-Plantard & Schneider（2003）的结果。还提供了随机抽样参数估计的应用。
Classical and non-classical central limit theorems for random sums of independent random variables of a double sequence
本文研究了二重序列独立随机变量随机和的经典与非经典中心极限定理，提出随机化方法并给出有效性条件，是对已知结果的扩展与推广。
Symmetric and centered binomial approximation of sums of locally dependent random variables
Stein方法用于用中心对称二项分布来近似离散和局部相依随机变量的和。在适当的光滑性质的被加数，在贝里-埃森定理的精度相同的顺序实现。本文还讨论了点过程的点总数的逼近问题。将结果应用于$r$ -扫描过程和Matérn硬核点过程I型的计算。
We provide a systematic approach to deal with the following problem. Let $X_1,\ldots,X_n$ be, possibly dependent, $[0,1]$-valued random variables. What is a sharp upper bound on the probability that their sum is significantly larger than their mean? In the case of independent random variables, a fundamental tool for bounding such probabilities is devised by Wassily Hoeffding. In this paper we consider analogues of Hoeffding's result for sums of dependent random variables for which we have certain information on their dependency structure. We prove a result that yields concentration inequalitie…
We obtain an uniform tail estimates for natural normed sums of independent random variables (r.v.) with regular varying tails of distributions. We give also many examples on order to show the exactness of offered estimates and discuss some applications in the method Monte-Carlo and statistics, and obtain the sufficient conditions for Central and stable limit theorem in the Banach space of continuous function. There are considered a slight generalization on a random variables with super-heavy tails and martingale difference scheme.
Localized large sums of random variables
本文研究了中心随机变量序列关于方差和的局部大部分和。给出了典型整数素因子分布的一个应用。
A comparison inequality for sums of independent random variables
本文给出了一个比较不等式，它可以用独立同分布随机变量的尾概率来估计独立巴拿赫空间值随机变量和的尾概率.更准确地说，设X_1，.，X_n是独立的Banach值随机变量。设I为独立于X_1的随机变量，.，X_n且均匀分布于{1，…，n}。设Z_1 = X_I，设Z_2，.，Z_n是Z_1的独立同分布拷贝。，则P（||X_1+...+ X_n||> t）< c P（||Z_1+...+ Z_n||> t/c），其中c是绝对常数。
Variance gamma approximation to sums of triplewise independent random variables
探讨三元独立随机变量与完全二分图的联系并构造其序列，用斯坦因方法对其和进行方差伽马近似并推导边界，还讨论了相关极限定理。
We consider moderately trimmed sums of non-negative i.i.d. random variables. We show that for every distribution function there exists a proper moderate trimming such that for the trimmed sum a non-trivial strong law of large numbers holds. In case that the distribution function has regularly varying tails we give necessary and sufficient conditions on the trimming for a strong law of large numbers to hold.
Convergence of random sums in non-commutative probability
研究非交换概率空间中自伴变量随机和的收敛性，通过建立*收敛结果，证明联合*收敛与停止随机变量矩及均值方差比收敛的等价性，得到多种变量随机和的中心极限定理及缩放*收敛极限。
Non-uniform estimates are obtained for Poisson, compound Poisson, translated Poisson, negative binomial and binomial approximations to sums of of m-dependent integer-valued random variables. Estimates for Wasserstein metric also follow easily from our results. The results are then exemplified by the approximation of Poisson binomial distribution, 2-runs and $m$-dependent $(k_1,k_2)$-events.
In this paper, we consider the sums of non-negative integer valued $m$-dependent random variables, and its approximation to the power series distribution. We first discuss some relevant results for power series distribution such as Stein operator, uniform and non-uniform bounds on the solution of Stein equation, and etc. Using Stein's method, we obtain the error bounds for the approximation problem considered. As special cases, we discuss two applications, namely, $2$-runs and $(k_1,k_2)$-runs and compare the bound with the existing bounds.
Asymptotic expansions for distributions of compound sums of light subexponential random variables
我们推导出一个渐近展开的复合和独立随机变量的分布，所有具有相同的轻尾次指数分布。泊松和几何数的被加数的例子作为说明的主要结果。对韦布尔分布进行了完整的计算，作为例子，我们毫无困难地导出了7项展开式。
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.
For a sequence $\{X_{n}, \, n \geqslant 1 \}$ of nonnegative random variables where $\max[\min(X_{n} - s,t),0]$, $t > s \geqslant 0$, satisfy a moment inequality, sufficient conditions are given under which $\sum_{k=1}^n (X_k - \mathbb{E} \, X_k)/b_n \overset{\mathrm{a.s.}}{\longrightarrow} 0$. Our statement allows us to obtain a strong law of large numbers for sequences of pairwise negatively quadrant dependent random variables under sharp normalising constants.
We present an entropy comparison result concerning weighted sums of independent and identically distributed random variables.
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…
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…
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