On L 1 Bounds for Asymptotic Normality of Some Weakly Dependent Random Variables

In the present paper, we consider L ₁ bounds for asymptotic normality for the sequence of r.v.’s X ₁,X ₂,… (not necessarily stationary) satisfying the ψ-mixing condition. The L ₁ bounds have been obtained in terms of Lyapunov fractions which, in a particular case, under finiteness of the third moments of summands and the finiteness of ∑ _r≥1 r ² ψ(r), are of order O(n ^−1/2), where the function ψ participates in the definition of the ψ-mixing condition.      

A Strong Law for Mixing Random Variables

A sharp almost sure bound is derived for limit points of average sum of weakly dependent random variables, which ensures strong laws of large numbers for and -mixing random variables, without assumptions on rate of tending to zero of and -mixing parameters _n and _n.     

On normal approximation of discounted and strongly mixing random variables

We estimate the difference for bounded functions h: ℝ → ℝ satisfying the Lipschitz condition, where Z _v = B _v ⁻¹ ∑ _i=0 ^∞ v ⁱ X _i and with discount factor ν such that 0 < ν < 1. Here {X _n , n ≥ 0} is a sequence of strongly mixing random variables with , and N is a standard normal random variable. In a particular case, the obtained upper bounds are of order O((1 − ν)^1/2). Published in Lietuvos Matematikos Rinkinys, Vol. 47, No. 3, pp. 399–409, July–September, 2007. The research was partially supported by the Lithuanian State Science and Studies Foundation, grant No. T-15/07.     

Normal Approximations for Descents and Inversions of Permutations of Multisets

Normal approximations for descents and inversions of permutations of the set {1,2,…,n} are well known. We consider the number of inversions of a permutation π(1),π(2),…,π(n) of a multiset with n elements, which is the number of pairs (i,j) with 1≤i<j≤n and π(i)>π(j). The number of descents is the number of i in the range 1≤i<n such that π(i)>π(i+1). We prove that, appropriately normalized, the distribution of both inversions and descents of a random permutation of the multiset approaches the normal distribution as n→∞, provided that the permutation is equally likely to be any possible permutation of the multiset and no element occurs more than α n times in the multiset for a fixed α with 0<α<1. Both normal approximation theorems are proved using the size bias version of Stein’s method of auxiliary randomization and are accompanied by error bounds. This work was supported by a research fellowship from the Sloan Foundation.     

The Central Limit Problem for Random Vectors with Symmetries

Motivated by the central limit problem for convex bodies, we study normal approximation of linear functionals of high-dimensional random vectors with various types of symmetries. In particular, we obtain results for distributions which are coordinatewise symmetric, uniform in a regular simplex, or spherically symmetric. Our proofs are based on Stein’s method of exchangeable pairs; as far as we know, this approach has not previously been used in convex geometry. The spherically symmetric case is treated by a variation of Stein’s method which is adapted for continuous symmetries. This work was done while at Stanford University.     

Strong Approximation Theorems for Sums of Random Variables when Extreme Terms are Excluded

Let {X _n ;n≥1} be a sequence of i.i.d. random variables and let X ^(r) _n = X _j if |X _j | is the r-th maximum of |X ₁|, ..., |X _n |. Let S _n = X ₁+⋯+X _n and ^(r) S _n = S _n −(X ⁽¹⁾ _n +⋯+X ^(r) _n ). Sufficient and necessary conditions for ^(r) S _n approximating to sums of independent normal random variables are obtained. Via approximation results, the convergence rates of the strong law of large numbers for ^(r) S _n are studied. Received March 22, 1999, Revised November 6, 2000, Accepted March 16, 2001     

On Quantiles and the Central Limit Question for Strongly Mixing Sequences

Three classes of strictly stationary, strongly mixing random sequences are constructed, in order to provide further information on the borderline of the central limit theorem for strictly stationary, strongly mixing random sequences. In these constructions, a key role is played by quantiles, as in a related construction of Doukhan et al.⁽¹¹⁾     

On the discounted global CLT for some weakly dependent random variables

In this paper, we consider L ₁ upper bounds in the global central limit theorem for the sequence of r.v.’s (not necessarily stationary) satisfying the ψ-mixing condition. In a particular case, under the finiteness of the third absolute moments of summands A _i and that of the series ∑ _r⩾1 r ² φ(r), we obtain bounds of order O(n ^−1/2) for Δ _n1:= ∫ _−∞ ^∞ |ℙ{A ₁ + ⋯ + A _n < x} − Φ(x)|dx, where is the standard normal distribution function, and ψ is the function participating in the definition of the ψ-mixing condition. Moreover, we apply the obtained results to get the convergence rate in the so-called discounted global CLT for a sequence of r.v.’s, satisfying the ψ-mixing condition. The bounds obtained provide convergence rates in the discounted global CLT of the same order as in the case of i.i.d. summands with a finite third absolute moment, i.e., of order O((1 − υ)^1/2), where υ is a discount factor, 0 < υ < 1. Published in Lietuvos Matematikos Rinkinys, Vol. 46, No. 4, pp. 584–597, October–December, 2006.     

Complete Convergence for Weighted Sums of Negatively Sup eradditive Dep endent Random Variables

In the paper, the complete convergence for the maximum of weighted sums of negatively superadditive dependent(NSD, in short) random variables is investigated by using the Rosenthal type inequality. Some su?cient conditions are presented to prove the complete convergence. The result obtained in the paper generalizes some corresponding ones for independent random variables and negatively associated random variables.     

On Complete Convergence for Arrays of Rowwise Strong Mixing Random Variables

In this paper, we present a general method to prove the complete conver- gence for arrays of rowwise strong mixing random variables, and give some results on complete convergence under some suitable conditions. Some Marcinkiewicz-Zygmund type strong laws of large numbers are also obtained.     

Lr Convergence for Arrays of Rowwise Negatively Sup eradditive Dep endent Random Variables

Let {Xnk, k≥1, n≥1} be an array of rowwise negatively superadditive depen-dent random variables and {an, n ≥ 1} be a sequence of positive real numbers such that an ↑ ∞. Under some suitable conditions, Lr convergence of a1n 1max≤j≤n ied. The results obtained in this paper generalize and improve some corresponding ones for negatively associated random variables and independent random variables. fl fl fl fl jP k=1 Xnk fl fl flfl is stud-ied. The results obtained in this paper generalize and improve some corresponding ones for negatively associated random variables and independent random variables.     

Maximal Inequalities and an Invariance Principle for a Class of Weakly Dependent Random Variables

The aim of this paper is to investigate the properties of the maximum of partial sums for a class of weakly dependent random variables which includes the instantaneous filters of a Gaussian sequence having a positive continuous spectral density. The results are used to obtain an invariance principle for strongly mixing sequences of random variables in the absence of stationarity or strong mixing rates. An additional condition is imposed to the coefficients of interlaced mixing. The results are applied to linear processes of strongly mixing sequences.     

NOD随机变量阵列加权乘积和的强极限定理

本文研究了行为NOD随机变量阵列加权乘积和的完全收敛性和强稳定性,推广和改进了文献[1]和[2]在NA情形时的结果以及[4]在独立同分布情形时的结果.     

On Random Attractors for Mixing Type Systems

The paper deals with infinite-dimensional random dynamical systems. Under the condition that the system in question is of mixing type and possesses a random compact attracting set, we show that the support of the unique invariant measure is the minimal random point attractor. The results obtained apply to the randomly forced 2D Navier–Stokes system.     

On Lower Bounds for Lp Norms in the Central Limit Theorem for Independent and m-Dependent Random Variables

We derive lower bounds for L_p norms , in the central limit theorem for independent and m–dependent random variables with finite fifth order absolute moments and for independent and m–dependent identically distributed random variables with fourth order moments.     

二维连续型随机变量相关计算的积分限确定问题

对二维连续型随机变量的积分定限问题进行了简要的归纳和总结,并给出了几类常见计算的基本方法和技巧,以期对学生的积分计算有所帮助.     

On the Central Limit Theorem and Its Weak Invariance Principle for Strongly Mixing Sequences with Values in a Hilbert Space via Martingale Approximation

In this paper we not only prove an extension to Hilbert spaces of a sharp central limit theorem for strongly real-valued mixing sequences, but also slightly improve it. The proof is mainly based on the Bernstein blocking technique and approximations by martingale differences. Moreover, we derive also the corresponding functional central limit theorem.     

On Some New Integral Inequalities for Functions in One and Two Variables

In this paper, we consider a bound on a general version of the integral inequalities for functions and also study the qualitative behavior of the solutions of certain classes of the hyperbolic partial delay differential equations under the integral inequalities.     

The Law of the Iterated Logarithm for Weakly Dependent Hilbert Space Valued Random Variables

We consider the bounded and compact laws of the iterated logarithm for weakly dependent Hilbert space valued random variables. Under optimal moment conditions, we prove the bounded and compact laws of the iterated logarithm for sequences of identically distributed Hilbert space valued random variables satisfying the uniform strong mixing condition.     

Precise Rates in the Law of Iterated Logarithm for the Moment of I.I.D. Random Variables

Let{X,Xn;n≥1} be a sequence of i,i.d, random variables, E X = 0, E X^2 = σ^2 〈 ∞.Set Sn=X1＋X2＋…＋Xn,Mn=max k≤n│Sk│,n≥1.Let an=O（1/loglogn）.In this paper,we prove that,for b〉-1,lim ε→0 →^2（b＋1）∑n=1^∞ （loglogn）^b/nlogn n^1/2 E{Mn-σ（ε＋an）√2nloglogn}＋σ2^-b/（b＋1）（2b＋3）E│N│^2b＋3∑k=0^∞ （-1）k/（2k＋1）^2b＋3 holds if and only if EX=0 and EX^2=σ^2〈∞.