PRECISE ASYMPTOTICS IN SELF-NORMALIZED SUMS OF ITERATED LOGARITHM FOR MULTIDIMENSIONALLY INDEXED RANDOM VARIABLES

In the case of Z＋^d（d ≥ 2）-the positive d-dimensional lattice points with partial ordering ≤, {Xk,k∈ Z＋^d} i.i.d, random variables with mean 0, Sn =∑k≤nXk and Vn^2 = ∑j≤nXj^2, the precise asymptotics for ∑n1/｜n｜（log｜n｜dP（｜Sn/Vn｜≥ε√log log｜n｜） and ∑n（logn｜）b/｜n｜（log｜n｜）^d-1P（｜Sn/Vn｜≥ε√log n）,as ε↓0,is established.

A Note on “Limit Distributions of Self-Normalized Sums” Using Cauchy-Generated Samples

In this case study, we would like to illustrate the utility of characteristic functions, using an example of a sample statistic defined for samples from Cauchy distribution. The derivation of the corresponding asymptotic probability density function is based on [1], elaborating and expanding the individual steps of their presentation, and including a small extension;our reason for such a plagiarism is to make the technique, its mathematical tools and ingenious arguments available to the widest possible audience.

Berry-Esseen bounds for self-normalized sums of locally dependent random variables

The Berry-Esseen bound provides an upper bound on the Kolmogorov distance between a random variable and the normal distribution.In this paper,we establish Berry-Esseen bounds with optimal rates for self-normalized sums of locally dependent random variables,assuming only a second-moment condition.Our proof leverages Stein's method and introduces a novel randomized concentration inequality,which may also be of independent interest for other applications.Our main results have applied to self-normalized sums of m-dependent random variables and graph dependency models.

A strong approximation of self-normalized sums

Let {X,Xn,n1} be a sequence of independent identically distributed random variables with EX=0 and assume that EX2I(|X|≤x) is slowly varying as x→∞,i.e.,X is in the domain of attraction of the normal law.In this paper a Strassen-type strong approximation is established for self-normalized sums of such random variables.

Complete Convergenceand Complete Moment Convergence for Weighted Sums of ANA Random Variables

In this paper,we investigate the complete convergence and complete moment conver-gence for weighted sums of arrays of rowwise asymptotically negatively associated(ANA)random variables,without assuming identical distribution.The obtained results not only extend those of An and Yuan[1]and Shen et al.[2]to the case of ANA random variables,but also partially improve them.

基于k-sums分段聚类的动态组合学习光伏短期功率预测

目前单一模型预测精度存在难以随着功率波动保持最优的问题,为提高并网系统运行的稳定性和电网的节能调度,文中提出了一种基于k-sums分层聚类的动态学习组合光伏短期功率预测方法。利用k-sums算法经过分段聚类,将天气类型分为晴天A 1、多云A 2、阴雨天B。通过TCN(Temporal Convolutional Network)提取数据的时序特征,并结合GRU(Gate Recurrent Unit)建立融合提取时序特征模块的改进GRU结构,以达到对时序特征敏感的效果。将改进GRU结构与SVM(Support Vector Machine)动态组合,使用Elastic Net算法输出最佳权重值叠加得到最终预测值。文中采用江苏某地区的光伏发电功率数据及对应的气象数据对所提方法进行验证,结果表明动态组合学习模型的MAE(Mean Absolute Error)为1.888,RMSE(Root Mean Squared Error)为2.403。

Ext-ICAS:A Novel Self-Normalized Extractive Intra Cosine Attention Similarity Summarization

With the continuous growth of online news articles,there arises the necessity for an efficient abstractive summarization technique for the problem of information overloading.Abstractive summarization is highly complex and requires a deeper understanding and proper reasoning to come up with its own summary outline.Abstractive summarization task is framed as seq2seq modeling.Existing seq2seq methods perform better on short sequences;however,for long sequences,the performance degrades due to high computation and hence a two-phase self-normalized deep neural document summarization model consisting of improvised extractive cosine normalization and seq2seq abstractive phases has been proposed in this paper.The novelty is to parallelize the sequence computation training by incorporating feed-forward,the self-normalized neural network in the Extractive phase using Intra Cosine Attention Similarity(Ext-ICAS)with sentence dependency position.Also,it does not require any normalization technique explicitly.Our proposed abstractive Bidirectional Long Short Term Memory(Bi-LSTM)encoder sequence model performs better than the Bidirectional Gated Recurrent Unit(Bi-GRU)encoder with minimum training loss and with fast convergence.The proposed model was evaluated on the Cable News Network(CNN)/Daily Mail dataset and an average rouge score of 0.435 was achieved also computational training in the extractive phase was reduced by 59%with an average number of similarity computations.

SELF-NORMALIZED CENTRAL LIMIT THEOREM AND ESTIMATION OF VARIANCE OF PARTIAL SUMS FOR NEGATIVE DEPENDENT RANDOM VARIABLES

Let ｛X n,n≥1｝ be a stationary LNQD or NA sequence satisfying EX 1=μ,EX 2 1<∞ and (Var S n)/n→σ 2 as n→∞.In this paper a class of self-normalized central limit theorems and estimators of Var S n are studied.The weak and strong consistency of the estimators of Var S n are presented.

A weak invariance principle for self-normalized products of sums of mixing sequences

Let variables in the {X, Xn, n ≥ 1} be a sequence of strictly stationary φ-mixing positive random domain of attraction of the normal law. Under some suitable conditions the principle for self-normalized products of partial sums is obtained.

基于Bernstein Copula函数的随机变量序列的Max-Sum局部等价式

该文考虑一类具有局部长尾分布,但不一定具有相同分布的随机变量序列,其联合分布由Bernstein copula函数进行联系.研究其部分和及其最大值的局部分布的渐近性质.在假设诸随机变量服从局部次指数分布的条件下,得到了Max-Sum局部等价性.该等价性从局部和相依的角度描述了随机游动的一次大跳原理.数值实验表明所得结果稳定可行.

Estimating Sums of Convergent Series via Rational Polynomials

Sums of convergent series for any desired number of terms, which may be infinite, are estimated very accurately by establishing definite rational polynomials. For infinite number of terms the sum infinite is obtained by taking the asymptotic limit of the rational polynomial. A rational function with second-degree polynomials both in the numerator and denominator is found to produce excellent results. Sums of series with different characteristics such as alternating signs are considered for testing the performance of the proposed approach.

Intelligent reflecting surface for sum rate enhancement in MIMO systems

The research for the Intelligent Reflecting Surface(IRS)which has the advantages of cost and energy efficiency has been studied.Channel capacity can be effectively increased by appropriately setting the phase value of IRS elements according to the channel conditions.However,the problem of obtaining an appropriate phase value of IRs is difficult to solve due to the non-convex problem.This paper proposes an iterative algorithm for the alternating optimal solution in the Single User Multiple-Input-Multiple-Output(SU-MIMO)systems.The proposed iterative algorithm finds an alternating optimal solution that is the phase value of IRS one by one.The results show that the proposed method has better performance than that of the randomized IRS systems.The number of iterations for maximizing the performance of the proposed algorithm depends on the channel state between the IRS and the receiver.

Large-scale self-normalizing neural networks

Self-normalizing neural networks(SNN)regulate the activation and gradient flows through activation functions with the self-normalization property.As SNNs do not rely on norms computed from minibatches,they are more friendly to data parallelism,kernel fusion,and emerging architectures such as ReRAM-based accelerators.However,existing SNNs have mainly demonstrated their effectiveness on toy datasets and fall short in accuracy when dealing with large-scale tasks like ImageNet.They lack the strong normalization,regularization,and expression power required for wider,deeper models and larger-scale tasks.To enhance the normalization strength,this paper introduces a comprehensive and practical definition of the self-normalization property in terms of the stability and attractiveness of the statistical fixed points.It is comprehensive as it jointly considers all the fixed points used by existing studies:the first and second moment of forward activation and the expected Frobenius norm of backward gradient.The practicality comes from the analytical equations provided by our paper to assess the stability and attractiveness of each fixed point,which are derived from theoretical analysis of the forward and backward signals.The proposed definition is applied to a meta activation function inspired by prior research,leading to a stronger self-normalizing activation function named‘‘bi-scaled exponential linear unit with backward standardized’’(bSELU-BSTD).We provide both theoretical and empirical evidence to show that it is superior to existing studies.To enhance the regularization and expression power,we further propose scaled-Mixup and channel-wise scale&shift.With these three techniques,our approach achieves 75.23%top-1 accuracy on the ImageNet with Conv MobileNet V1,surpassing the performance of existing self-normalizing activation functions.To the best of our knowledge,this is the first SNN that achieves comparable accuracy to batch normalization on ImageNet.

基于BCUSUM的多参数变点估计

文章基于递归残差的逆序特征和隔离检测研究了回归模型多参数变点的检测方法。首先,构建带有变点的回归模型,考虑到多元正向CUSUM检验能防止协变量均值与偏移量正交时损失功效,但其变点检测效果并不理想的情况,引入修正的检验统计量BCUSUM。其次,结合快速高效的隔离检测技术,提出MCPDP算法用于估计变点数目及位置。最后,模拟结果表明,所提出的方法能较好地控制检验水平,有更高的功效;评价结果显示,MCPDP算法在变点估计性能方面表现较优;实例分析表明,交通流变点符合实际交通情况,验证了该方法的有效性,且所构建的模型可以作为交通参数确定性经验关系的一种修正。

A vector sum analysis method for stability evolution of expansive soil slope considering shear zone damage softening

Slope stability analysis is a classical mechanical problem in geotechnical engineering and engineering geology.It is of great significance to study the stability evolution of expansive soil slopes for engineering construction in expansive soil areas.Most of the existing studies evaluate the slope stability by analyzing the limit equilibrium state of the slope,and the analysis method for the stability evolution considering the damage softening of the shear zone is lacking.In this study,the large deformation shear mechanical behavior of expansive soil was investigated by ring shear test.The damage softening characteristic of expansive soil in the shear zone was analyzed,and a shear damage model reflecting the damage softening behavior of expansive soil was derived based on the damage theory.Finally,by skillfully combining the vector sum method and the shear damage model,an analysis method for the stability evolution of the expansive soil slope considering the shear zone damage softening was proposed.The results show that the shear zone subjected to large displacement shear deformation exhibits an obvious damage softening phenomenon.The damage variable equation based on the logistic function can be well used to describe the shear damage characteristics of expansive soil,and the proposed shear damage model is in good agreement with the ring shear test results.The vector sum method considering the damage softening behavior of the shear zone can be well applied to analyze the stability evolution characteristics of the expansive soil slope.The stability factor of the expansive soil slope decreases with the increase of shear displacement,showing an obvious progressive failure behavior.

Study on Effects of Short Circuit Between High and Common Ends of LVDT's Lead Wires on Sum Voltage

The Linear variable differential transformer(LVDT)is widely used in the aircraft field and crucial to the safety of the aircraft.Based on the short-circuit failure mode between the high and common ends of the LVDT′s lead wires,the theoretical formula of short-circuit sum voltage was deduced and verified via tests.Moreover,the effects of the core length and the winding resistance on the short-circuit sum voltage were analyzed and discussed through numerical methods.The present research put forward some references for LVDT′s design.

Nash Equilibrium of a Fixed-Sum Two-Player Game

It is well established that Nash equilibrium exists within the framework of mixed strategies in strategic-form non-cooperative games. However, finding the Nash equilibrium generally belongs to the class of problems known as PPAD (Polynomial Parity Argument on Directed graphs), for which no polynomial-time solution methods are known, even for two-player games. This paper demonstrates that in fixed-sum two-player games (including zero-sum games), the Nash equilibrium forms a convex set, and has a unique expected payoff. Furthermore, these equilibria are Pareto optimal. Additionally, it is shown that the Nash equilibrium of fixed-sum two-player games can theoretically be found in polynomial time using the principal-dual interior point method, a solution method of linear programming.

Precise large deviation result for heavy-tailed random sums and applications to risk theory

The differences between two sequences of nonnegative independent and identically distributed random variables with sub-exponential tails and the random index are studied. The random index is a strictly stationary renewal counting process generated by some negatively associated random variables. Using a revised large deviation result of partial sums, the elementary renewal theorem and the central limit theorem of negatively associated random variables, a precise large deviation result is derived for the random sums. The result is applied to the customer-arrival-based insurance risk model. Some uniform asymptotics for the ruin probabilities of an insurance company are obtained as the number of customers or the time tends to infinity.

Almost sure central limit theory for products of sums of partial sums

Consider a sequence of i.i.d.positive random variables.An universal result in almost sure limit theorem for products of sums of partial sums is established.We will show that the almost sure limit theorem holds under a fairly general condition on the weight dk= k-1 exp（lnβk）,0≤β〈1.And in a sense,our results have reached the optimal form.

ON THE LIMITING BEHAVIOR OF THE MAXIMUM PARTIAL SUMS FOR ARRAYS OF ROWWISE NA RANDOM VARIABLES

Let {Xni, 1 ≤ n,i 〈 ∞} be an an array of rowwise NA random variables and {an, n ≥ 1} a sequence of constants with 0 〈 an ↑∞ . The limiting behavior of maximum partial sums 1/an max 1≤k≤n｜^k∑i=1 Xni｜ is investigated and some new results are obtained. The results extend and improve the corresponding theorems of rowwise independent random variable arrays by Hu and Taylor [1] and Hu and Chang [2].