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 self-normalized Dickey-Fuller test for unit root in autoregressive time series with GARCH errors

In this article, the unit root test for AR（p） model with GARCH errors is considered. The Dickey-Fuller test statistics are rewritten in the form of self-normalized sums, and the asymptotic distribution of the test statistics is derived under the weak conditions.

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.

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.

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.

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.

Normalized and self-normalized Cramér-type moderate deviations for the Euler-Maruyama scheme for the SDE

In this paper,we establish normalized and self-normalized Cramér-type moderate deviations for the Euler-Maruyama scheme for SDE.Due to our results,Berry-Esseen's bounds and moderate deviation principles are also obtained.Our normalized Cramér-type moderate deviations refine the recent work of Lu et al.(2022).

Self-normalized moderate deviations for independent random variables

Let X1,X2,... be a sequence of independent random variables （r.v.s） belonging to the domain of attraction of a normal or stable law. In this paper, we study moderate deviations for the self-normalized sum n X ∑^n_i=1Xi/Vm,p ,where Vn,p （∑^n_i=1｜Xi｜p）^1/p （P 〉 1）.Applications to the self-normalized law of the iteratedlogarithm, Studentized increments of partial sums, t-statistic, and weighted sum of independent and identically distributed （i.i.d.） r.v.s are considered.

Berry-Esseen Bounds for Self-Normalized Martingales

A Berry–Esseen bound is obtained for self-normalized martingales under the assumption of finite moments.The bound coincides with the classical Berry–Esseenboundforstandardizedmartingales.Anexampleisgiventoshowtheoptimality of the bound.Applications to Student’s statistic and autoregressive process are also discussed.

Self-normalized Cramer-type Moderate Deviations for Functionals of Markov Chain

Let{xn,n≥0}be a Markov chain with a countable state space S and let f(·)be a measurable function from S to R and consider the functionals of the Markov chain yn:=f(xn).We construct a new type of self-normalized sums based on the random-block scheme and establish a Crame′r-type moderate deviations for self-normalized sums of functionals of the Markov chain.

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.

An Almost Sure Central Limit Theorem for Self-normalized Weighted Sums

Let X, X1, X2, be a sequence of nondegenerate i.i.d, random variables with zero means, which is in the domain of attraction of the normal law. Let （ani, 1 ≤ i ≤n,n ≥1} be an array of real numbers with some suitable conditions. In this paper, we show that a central limit theorem for self-normalized weighted sums holds. We also deduce a version of ASCLT for self-normalized weighted sums.

On necessary and sufficient conditions for the self-normalized central limit theorem In Honor of Professor Chuanrong Lu on His 85th Birthday

Let X_1, X_2,... be a sequence of independent random variables and S_n=sum X_1 from i=1 to n and V_n^2=sum X_1~2 from i=1 to n . When the elements of the sequence are i.i.d., it is known that the self-normalized sum S_n/V_n converges to a standard normal distribution if and only if max1≤i≤n|X_i|/V_n → 0 in probability and the mean of X_1 is zero. In this paper, sufficient conditions for the self-normalized central limit theorem are obtained for general independent random variables. It is also shown that if max1≤i≤n|X_i|/V_n → 0 in probability, then these sufficient conditions are necessary.

Self-Normalized Moderate Deviation and Laws of the Iterated Logarithm Under G-Expectation

The sub-linear expectation or called G-expectation is a non-linear expectation having advantage of modeling non-additive probability problems and the volatilityuncertainty in finance.Let{Xn;n≥1}be a sequence of independent random vari-ables in a sub-linear expectation space(Ω,H,E^(^)).Denote S_(n)=∑_(k=1)^(n)Xk and=V_(n)^(2)=∑_(k=1)^(n)X_(k)^(2).In this paper,a moderate deviation for self-normalized sums,thatis,the asymptotic capacity of the event{Sn/Vn≥x_(n)}for x_(n)=o(√n),is found both for identically distributed random variables and independent but not necessarilyidentically distributed random variables.As an application,the self-normalized lawsof the iterated logarithm are obtained.A Bernstein's type inequality is also establishedfor proving the law of the iterated logarithm.

Convergence to a self-normalized G-Brownian motion

G-Brownian motion has a very rich and interesting new structure that nontrivially generalizes the classical Brownian motion.Its quadratic variation process is also a continuous process with independent and stationary increments.We prove a self-normalized functional central limit theorem for independent and identically distributed random variables under the sub-linear expectation with the limit process being a G-Brownian motion self-normalized by its quadratic variation.To prove the self-normalized central limit theorem,we also establish a new Donsker’s invariance principle with the limit process being a generalized G-Brownian motion.

A Self-normalized Law of the Iterated Logarithm for the Geometrically Weighted Random Series

Let {X, Xn; n ≥ 0} be a sequence of independent and identically distributed random variables with EX=0, and assume that EX^2I（｜X｜ ≤ x） is slowly varying as x →∞, i.e., X is in the domain of attraction of the normal law. In this paper, a self-normalized law of the iterated logarithm for the geometrically weighted random series Σ~∞（n=0）β~nXn（0 〈 β 〈 1） is obtained, under some minimal conditions.

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.

Self-normalization:Taming a wild population in a heavy-tailed world

The past two decades have witnessed the active development of a rich probability theory of Studentized statistics or self-normalized processes, typified by Student’s t-statistic as introduced by W. S. Gosset more than a century ago, and their applications to statistical problems in high dimensions, including feature selection and ranking, large-scale multiple testing and sparse, high dimensional signal detection. Many of these applications rely on the robustness property of Studentization/self-normalization against heavy-tailed sampling distributions. This paper gives an overview of the salient progress of self-normalized limit theory, from Student’s t-statistic to more general Studentized nonlinear statistics. Prototypical examples include Studentized one- and two-sample U-statistics. Furthermore, we go beyond independence and glimpse some very recent advances in self-normalized moderate deviations under dependence.

Saddlepoint approximations for studentized compound Poisson sums with no moment conditions in audit sampling

Saddlepoint approximations for the studentized compound Poisson sums with no moment conditions in audit sampling are derived. This result not only provides a very accurate approximation for studentized compound Poisson sums, but also can be applied much more widely in statistical inference of the error amount in an audit population of accounts to check the validity of financial statements of a firm. Some numerical illustrations and comparison with the normal approximation method are presented.

Studentized Increments of Partial Sums

Using suitable self-normalization for partial sums of i.i.d.random variables,Griffin and Kuelbs established the law of the iterated logarithm for all distributions in the domain of attraction of a normal law.We obtain the corresponding results for Studentized increments of partial sums under thesame condition.