B值随机变量阵列加权和的完全收敛性

设{X_ni:1≤i≤n,n≥1}为行间独立的B值r.v.阵列,g(z)是指数为1/p的正则变化函数,r>0,{ani 1≤t≤n,n≥1}为实数阵列,本文得到了使(?)成立的条件,推广并改进了Stout及Sung等的著名结论.     

NA随机变量加权和的收敛性

利用Rosenthal型最大值不等式,得到了NA随机变量加权和的Marcinkiewicz-Zygmund强大数定律和完全收敛性,所获结果推广和改进了一些文献中相应的结果.     

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〈∞.     

$\rho^*$-混合随机变量列加权和的完全收敛性

In this paper, the complete convergence of weighted sums for ρ*-mixing sequence of random variables is investigated. By applying moment inequality and truncation methods, the equivalent conditions of complete convergence of weighted sums for ρ*-mixing sequence of random variables are established. We not only promote and improve the results of Li et al. (J. Theoret. Probab., 1995, 8(1): 49-76) from i.i.d. to ρ*-mixing setting but also obtain their necessities and relax their conditions.     

I.I.D.随机变量序列矩完全收敛的精确渐近性

{X,Xn;n≥1}为独立同分布的随机变量序列, EX=0,01 p/2满足E|X|r<∞,且E|X|3<∞,那么其中Z服从均值为0,方差为σ2的正态分布.     

On the Rate of Decay of the Concentration Function of the Sum of Independent Random Variables

Let X₁, ... , X_n be i.i.d. integral valued random variables and S_n their sum. In the case when X₁ has a moderately large tail of distribution, Deshouillers, Freiman and Yudin gave a uniform upper bound for max _{k ∊ ℤ} Pr{S_n = k} (which can be expressed in term of the Lévy Doeblin concentration of S_n), under the extra condition that X₁ is not essentially supported by an arithmetic progression. The first aim of the paper is to show that this extra condition cannot be simply ruled out. Secondly, it is shown that if X₁ has a very large tail (larger than a Cauchy-type distribution), then the extra arithmetic condition is not sufficient to guarantee a uniform upper bound for the decay of the concentration of the sum S_n. Proofs are constructive and enhance the connection between additive number theory and probability theory.À Jean-Louis Nicolas, avec amitié et respect2000 Mathematics Subject Classification: Primary—60Fxx, 60Exx, 11Pxx, 11B25     

Lim sup behavior of sums of geometrically weighted i.i.d. random variables

Geometrically weighted i.i.d. random variables {Y_n} which are bounded above are shown to exhibit iterated logarithm type behavior. Specifically, if b > 1 and if the lower tail of the distribution of Y₁ approaches 0 fast enough, then lim sup_n→∞(b?1) Σⁿ_j=1b¹Y_j?bⁿ⁺¹=L, almost certainly, where L is the essential supremum of Y₁.     

On the Relationship Between the Baum-Katz-Spitzer Complete Convergence Theorem and the Law of the Iterated Logarithm

For a sequence of i.i.d. Banach space-valued random variables {Xn; n ≥ 1} and a sequence of positive constants {an; n ≥ 1}, the relationship between the Baum-Katz-Spitzer complete convergence theorem and the law of the iterated logarithm is investigated. Sets of conditions are provided under which （i） lim sup n→∞ ｜｜Sn｜｜/an〈∞ a.s.and ∞ ∑n=1（1/n）P（｜｜Sn｜｜/an ≥ε〈∞for all ε 〉 λ for some constant λ ∈ [0, ∞） are equivalent; （ii） For all constants λ ∈ [0, ∞）, lim sup ｜｜Sn｜｜/an =λ a.s.and ^∞∑ n=1（1/n） P（｜｜Sn｜｜/an ≥ε）{〈∞, if ε〉λ =∞,if ε〈λare equivalent. In general, no geometric conditions are imposed on the underlying Banach space. Corollaries are presented and new results are obtained even in the case of real-valued random variables.     

Almost Sure Convergence of the General Jamison Weighted Sum of $$ {\user1{B}} $$-Valued Random Variables

In this paper．two new functions are introduced to depict the Jamison weighted sum of random variables instead using the common methods．their properties and relationships are systematically discussed．We also analysed the implication of the conditions in previous papers．Then we apply these consequences to B-valued random variables．and greatly improve the original results of the strong convergence of the general Janfison weighted sum．Furthermore,our discussions are useful to the corresponding questions of real-valued random variables．