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.     

On the limiting behavior of the maximum partial sums for arrays of rowwise NA random variables

Let {Xni, 1 ≤ n,i ＜∞} be 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| kΣi=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].     

ON CHOVER''''S LIL FOR THE WEIGHTED SUMS OF STABLE RANDOM VARIABLES

Let {Xn,n ≥ 1} be a sequence of α-stable random variables(0 < α < 2), {ani,1 ≤ i≤ n, n≥1} be an array of constant real numbers. Under some restriction of {ani,1 ≤ i ≤ n,n≥1}, the authors discuss the integral test for the weighted partial sums {Σi=1naniXi,n ≥ 1}, and obtain the Chover's laws of iterated logarithm(LIL) as corollaries.     

Strong Convergence for Weighted Sums of Negatively Associated Arrays

Let {Xni} be an array of rowwise negatively associated random variables and Tnk=k∑i=1 i^a Xni for a ≥ -1, Snk =∑｜i｜≤k Ф（i/nη）1/nη Xni for η∈（0,1],where Ф is some function. The author studies necessary and sufficient conditions of ∞∑n=1 AnP（max 1≤k≤n｜Tnk｜〉εBn）〈∞ and ∞∑n=1 CnP（max 0≤k≤mn｜Snk｜〉εDn）〈∞ for all ε 〉 0, where An, Bn, Cn and Dn are some positive constants, mn ∈ N with mn /nη →∞. The results of Lanzinger and Stadtmfiller in 2003 are extended from the i.i.d, case to the case of the negatively associated, not necessarily identically distributed random variables. Also, the result of Pruss in 2003 on independent variables reduces to a special case of the present paper; furthermore, the necessity part of his result is complemented.     

On the convergence for PNQD sequences with general moment conditions

Let {X,X_n,n≥ 1} be a sequence of identically distributed pairwise negative quadrant dependent(PNQD) random variables and {a_n,n≥ 1} be a sequence of positive constants with a_n=f(n) and f(θ~k)/f(θ~(k-1)≥β for all large positive integers k, where 1 θ≤β and f(x) 0(x≥1) is a non-decreasing function on [b,+∞) for some b≥1.In this paper,we obtain the strong law of large numbers and complete convergence for the sequence {X,X_n, n≥ 1},which are equivalent to the general moment condition∑_(n=1)~∞ P(|X| a_n) ∞.Our results extend and improve the related known works in Baum and Katz [1],Chen at al.[3],and Sung[14].     

Large deviation results for generalized compound negative binomial risk models

In this paper we extend and improve some results of the large deviation for random sums of random variables. Let {Xn;n 〉 1} be a sequence of non-negative, independent and identically distributed random variables with common heavy-tailed distribution function F and finite mean μ ∈R^＋, {N（n）; n ≥0} be a sequence of negative binomial distributed random variables with a parameter p C （0, 1）, n ≥ 0, let {M（n）; n ≥ 0} be a Poisson process with intensity λ 〉 0. Suppose {N（n）; n ≥ 0}, {Xn; n≥1} and {M（n）; n ≥ 0} are mutually independent. Write S（n） =N（n）∑i=1 Xi-cM（n）.Under the assumption F ∈ C, we prove some large deviation results. These results can be applied to certain problems in insurance and finance.     

负相伴随机变量的非经典重对数律

§ 1　 IntroductionA finite family of random variables { Xi,1≤ i≤ n} is said to be negatively associated(NA) is for every pair of disjointsubsets A1 and A2 of{ 1 ,2 ,...,n} ,Cov{ f1 (Xi,i∈ A1 ) ,f2 (Xj,j∈ A2 ) }≤ 0 ,(1 .1 )whenever f1 and f2 are coordinatewise increasing and the covariance exists.An infinitefamily is negatively associated ifevery finite subfamily is negatively associated.This defini-tion was introduced by Alam and Saxena[1 ] and Joag-Dev and Proschan[2 ] .As pointed…     

负相依随机变量的自正则化中心极限定理与部分和的方差估计

§ 1　 IntroductionWe firstintroduce some concepts.Random variables X and Y are called negative dependent ( ND) if for any pair ofmonotonically non-decresing functions f and g,Cov{ f( X) ,g( Y) }≤ 0 .Clearly itis equivalenttoP( X≤ x,Y≤ y)≤ P( X≤ x) P( Y≤ y)for all x,y∈R.A random sequence{ Xi,i≥ 1 } is said to be negative quadrant dependent( NQD) if any pairof variables Xi,Xj( i≠j) are ND.A sequence of random variables{ Xi,i≥ 1 } is said to be linear negative quadrand depend…     

Some Convergence Properties for Weighted Sums of Martingale Difference Random Vectors

Let {X_ni,F_ni;1≤i≤n,n≥1} be an array of R^d martingale difference random vectors and {A_ni,1≤i≤n,n≥1} be an array of m×d matrices of real numbers.In this paper,the Marcinkiewicz-Zygmund type weak law of large numbers for maximal weighted sums of martingale difference random vectors is obtained with not necessarily finite p-th(1相似文献   

负相协随机变量非随机和的差的精确大偏差

In this paper,we study precise large deviation for the non-random difference sum from j=1 to n_1(t) X_(1j)-sum from j=1 to n_2(t) X_(2j),where sum from j=1 to n_1(t) X_(1j) is the non-random sum of {X_(1j),j≥1} which is a sequence of negatively associated random variables with common distribution F_1(x),and sum from j=1 to n_2(t) X_(2j) is the non-random sum of {X_(2j),j≥1} which is a sequence of independent and identically distributed random variables,n_1(t) and n_2(t) are two positive integer functions.Under some other mild conditions,we establish the following uniformly asymptotic relation lim t→∞ sup x≥r(n_1(t))~(p+1)|(P(∑~(n_1(t)_(j=1)X_(1j)-∑~(n_2(t)_(j=1)X_(2j)-(μ_1n_1(t)-μ_2n_2(t)x))/(n_1(t)F_1(x))-1|=0.     

Asymptotics for the Tail Probability of Random Sums with a Heavy-Tailed Random Number and Extended Negatively Dependent Summands

Let {X,X_k:k≥1} be a sequence of extended negatively dependent random variables with a common distribution F satisfying EX 0.Let r be a nonnegative integer-valued random variable,independent of {X,X_k:k≥1}.In this paper,the authors obtain the necessary and sufficient conditions for the random sums S_r =(?)X_n to have a consistently varying tail when the random number t has a heavier tail than the summands,i.e.,(P(Xx))/(P(r x))→0as x→∞     

Some Convergence Sequences under Results for Arbitrary Moment Condition

Let {X n , n ≥ 1} be an arbitrary sequence of random variables. Some convergence results for the partial sums of arbitrary sequence of random variables are obtained, which generalize the known results for independent sequences, NA sequences, ρ-mixing sequences and φ-mixing sequences, and so on.     

ρ混合序列重对数律的精确率

§ 1　 Introduction and main resultsL et { X,Xn;n≥ 1} be a sequence of random variables with common distributionfunction F,mean0 and positive,finite variance,and set Sn= nk=1 Xk,n≥ 1.Also letlogx= ln(x∨e) ,log logx=log(logx) and(x) =2 xlog logx.Gut and Sp taru[2 ] studied theprecise asymptotics on the law of the iterated logarithm.One of their results is as follows.Theorem A.Spuuose that{ X ,Xn;n≥ 1} is a sequence of i.i.d.random variables with EX= 0 and0 相似文献   

Complete moment and integral convergence for sums of negatively associated random variables

For a sequence of identically distributed negatively associated random variables {Xn; n ≥ 1} with partial sums Sn = ∑i=1^n Xi, n ≥ 1, refinements are presented of the classical Baum-Katz and Lai complete convergence theorems. More specifically, necessary and sufficient moment conditions are provided for complete moment convergence of the form ∑n≥n0 n^r-2-1/pq anE（max1≤k≤n｜Sk｜^1/q-∈bn^1/qp）^＋〈∞to hold where r 〉 1, q 〉 0 and either n0 = 1,0 〈 p 〈 2, an = 1,bn = n or n0 = 3,p = 2, an = 1 （log n） ^1/2q, bn=n log n. These results extend results of Chow and of Li and Spataru from the indepen- dent and identically distributed case to the identically distributed negatively associated setting. The complete moment convergence is also shown to be equivalent to a form of complete integral convergence.     

A Supplement to the Baum-Katz-Spitzer Complete Convergence Theorem

Let {X, Xn; n≥ 1} be a sequence of i.i.d. Banach space valued random variables and let {an; n ≥ 1} be a sequence of positive constants such that an↑∞ and 1〈 lim inf n→∞ a2n/an≤lim sup n→∞ a2n/an〈∞ Set Sn=∑i=1^n Xi,n≥1.In this paper we prove that ∑n≥1 1/n P（｜｜Sn｜｜≥εan）〈∞ for all ε〉0 if and only if lim n→∞ Sn/an=0 a.s. This result generalizes the Baum-Katz-Spitzer complete convergence theorem. Combining our result and a corollary of Einmahl and Li, we solve a conjecture posed by Gut.     

Precise asymptotics in self-normalized sums of iterated logarithm for multidimensionally indexed random variables

In the case of Zd (d ≥ 2)-the positive d-dimensional lattice points with partial ordering ≤, {Xk,k ∈ Zd } i.i.d. random variables with mean 0, Sn = ∑k≤nXk and Vn2 = ∑j≤nX2j, the precise asymptotics for ∑n1/|n|(log|n|)dP(|Sn/vn|≥ ε√loglog|n|) and ∑n(logn|)δ/|n|(log|n|)d-1 P(|Sn/Vn| ≥ ε√log n), as ε ↘ 0, is established.     

Some Estimation Problems for Linear Model of Time Series,

If{X(n),n≥1}are random variables which satisfy following linear modelwhere α_j,λ_j,l≤j≤n are unknown constants andis real stationary sequence.In thispaper we discuss the estimation of λ_j,α_j and introduce δ-separated periodgram maximum estimatorλ_j for λ_j.Under some moderate conditions we prove the following consistence and asymptotionormality for these estimators,as the size     

Large deviations for sums of independent random variables with dominatedly varying tails

In this paper the large deviation results for partial and random sums Sn-ESn=n∑i=1Xi-n∑i=1EXi,n≥1;S(t)-ES(t)=N(t)∑i=1Xi-E(N(t)∑i=1Xi),t≥0are proved, where {N(t); t≥ 0} is a counting process of non-negative integer-valued random variables, and {Xn; n ≥ 1} are a sequence of independent non-negative random variables independent of {N(t); t ≥ 0}. These results extend and improve some known conclusions.     

On the strong convergence properties for weighted sums of negatively orthant dependent random variables

In the paper, the strong convergence properties for two different weighted sums of negatively orthant dependent(NOD) random variables are investigated. Let {X_n, n ≥ 1}be a sequence of NOD random variables. The results obtained in the paper generalize the corresponding ones for i.i.d. random variables and identically distributed NA random variables to the case of NOD random variables, which are stochastically dominated by a random variable X. As a byproduct, the Marcinkiewicz-Zygmund type strong law of large numbers for NOD random variables is also obtained.     

广义复合二项风险模型的若干大偏差结果

This paper is a further investigation of large deviation for partial and random sums of random variables, where {Xn,n ≥ 1} is non-negative independent identically distributed random variables with a common heavy-tailed distribution function F on the real line R and finite mean μ∈ R. {N（n）,n ≥ 0} is a binomial process with a parameter p ∈ （0,1） and independent of {Xn,n ≥ 1}; {M（n）,n ≥ 0} is a Poisson process with intensity λ 〉 0, Sn = ΣNn i=1 Xi-cM（n）. Suppose F ∈ C, we futher extend and improve some large deviation results. These results can apply to certain problems in insurance and finance.