Characterization of type<Emphasis Type="Italic">p</Emphasis> Banach spaces by the weak law of large numbers

For weighted sums of the form where {a _nj , 1 ⩽j⩽k _n ↑∞,n⩾1} is a real constant array and {X _aj , 1≤j≤k _{n, n}≥1} is a rowwise independent, zero mean, random element array in a real separable Banach space of typep, we establishL ^r convergence theorem and a general weak law of large numbers respectively, conversely, we characterize Banach spaces of typep in terms of convergence inr-th mean and probability for such weighted sums. Foundation item: Supported by the National Natural Science Foundation of China (No. 10071058) Biography: Gan Shi-xin (1939-), male, Professor, research direction: martingale theory, probability limiting theory and Banach space geometry theory.     

Exponential inequality for a class of NOD random variables and its application

In this paper,an exponential inequality for weighted sums of identically distributed NOD (negatively orthant dependent) random variables is established,by which we obtain the almost sure convergence rate of which reaches the available one for independent random variables in terms of Berstein type inequality. As application,we obtain the relevant exponential inequality for Priestley-Chao estimator of nonparametric regression estimate under NOD samples,from which the strong consistency rate is also obtained.     

LPQD列生成线性过程部分和的精确渐近性

设{ε_t;t∈Z^{+}是一严平稳零均值的LPQD随机变量序列, 并且02₁<∞, σ2, 0<σ2<∞, {a_j; j∈N }是一实数序列, 定义线性过程X_t. 利用弱收敛定理和矩不等式, 对一般的拟权函数和边界函数, 证明了{M_n}和{S_n}的精确渐近性.}     

On the limiting behavior of weighted partial sums for<Emphasis Type="Italic">B</Emphasis> valued martingale difference sequences

Let {X_n, n≥1} be a martingale difference sequence and {a _nk , 1⩽k⩽n,n⩾1} an array of constant real numbers. The limiting behavior of weighted partial sums ∑ _k=1 ⁿ a _nk X _k is investigated and some new results are obtained. Foundation item: Supported by the National Natural Sciene Foundation of China (No. 10071058 and No. 10071019) Biography: Gan Shi-xin (1939-), male, Prof lessor, research direction: martingale theory, probability limiting theory and Banach space geometry theory.     

Strong stability of linear forms in φ -mixing random variables

In this paper some new results of strong stability of linear forms in φ-mixing random variables are given. It is mainly proved that for a sequence of φ-mixing random variables {x _n , n ⩾ 1} and two sequences of positive numbers {a _n , n ⩾ 1} and {b _n , n ⩾ 1} there exist d _n ∈ R, n = 1,2,L, such that a.s. under some suitable conditions. The results extend and improve the corresponding theorems for independent identically distributed random variables. Foundation item: Supported by the National Natural Science Foundation of China(10671149)     

Multi-parameter infinite-dimensional (r,δ)-OU process

The definition of n-parameter infinite-dimensional (r,δ) -Ornstein-Uhlenbeck process {xt (·) } ((r, δ)-OUPn∞ for short) is given. The absolute continuity of distribution μt of Xt(·) and the limit of Xt(·) when |t| →∞ is discussed.     

Finite Time Ruin Probability with Variable Interest Rate and Extended Regular Variation

Consider an insurance risk model, in which the surplus process satisfies a recursive equation Un=Un-1(1 rn)-Xn for n≥1, where U0=x≥0 is the initial surplus, {rn；n≥1} the interest rate sequence, {Xn；n≥l} the sequence of i. i. d. real-valued random variables with common distribution function F, which denotes the gross loss during the nth year. We investigate the ruin probability within a finite time horizon and give the asymptotic result as x→∞.     

The Hàjek-Rènyi inequlity for Banach space valued random variable sequences and its application

In this paper we prove the following Hàjek-Rènyi inequality: Let 0≤p≤1, then for any Banach spaceB, anyL ^p integrableB valued random variable sequence {D _n , n≥1}, any real number sequence {b _n , n≥1} with 0<b _n, ↑ ∞, any integern≥1, there exist a constantC=C _p>0 (only depending onp) such that

Model of Markov Chains in Space-Time Random Environments

A general framework of stochastic model for a Markov chain in a space-time random environment is introduced, here the environment ξ^＊：={ξ1,x∈N,x∈ X}is a random field. We study the dependence relations between the environment and the original chain, especially the ＂feedback＂. Some equivalence theorems and law of large numbers are obtained.     

Almost sure convergence of weighted sums of NA sequences

For double arrays of constants {a _ni, 1≤i≤k _n, n≥1} and NA r.v. 's {X _n, n≥1}, conditions for almost sure convergence of are given. Both casesk _n ↑ ∞ andk _n=∞ are treated. A Marcinkiewicz-type theorem for i. d. NA sequences is obtained as a special case. Supported by the National Natural Science Foundation of China Cheng Riyan: born in 1968, MS student     

Almost sure convergence of weighted sums of NA sequences

For double arrays of constants {a _ni, 1≤i≤k _n, n≥1} and NA r.v. 's {X _n, n≥1}, conditions for almost sure convergence of are given. Both casesk _n ↑ ∞ andk _n=∞ are treated. A Marcinkiewicz-type theorem for i. d. NA sequences is obtained as a special case. Supported by the National Natural Science Foundation of China Cheng Riyan: born in 1968, MS student     

关于任意同分布随机变量序列最大值不等式及应用

设{X,Xn,n≥1}是同分布的随机变量序列(不必独立),记部分和Sn=∑ni=1Xi,n≥1。获得了max1≤k≤n︱Sk︱/n1/p的尾概率的一个上界,其中0p1。作为一个应用,给出了正则和极大值函数sup n≥1︱Sk︱/n1/p的r(r0)阶矩存在的充分条件,推广了独立情形相应的结果。     

B-Valued Dyadic Derivative

The principles of the new maximal operator H＊ we defined are discussed. We prove that it is bounded from martingale Hardy-Lorentz L^Xp.q[0,1） to the Lorentz L^Xp.q[0,1） for 1/2〈 p〈∞, 0〈~ q ≤ ∞, where X is any Banach space. When the Banach space X has the RN property, the sequence dnHnf converges to f a.e. Meanwhile the convergence in L^Xp norm for 1≤p〈∞ is a consequence of that the family functions K （n∈N） is an approximate identity.     

Absolute continuity of interacting measure-valued branching processes and its occupation-time processes

Let X t be the interaction measured_valued branching α_ symmetric stable process over R d(1<α≤2) constructed by Meleard_Roelly . Frist, it is shown that X t is absolutely continuous with respect to the Lebesgue measure (on R ) with a continuous density function which satisfies some SPDE. Second, it is proved that if the underlying process is a Brownian motion on R d (d≤3), the corresponding occupation_time process Y t is also absolutely continuous with respect to the Lebesgue measure.     

Some polar sets for the generalized Brownian sheet

Let (t)(t ∈ R ₊ ^N ) be the d-dimensional N-parameter generalized Brownian sheet. We study the polar sets for (t). It is proved that for any a ∈ R ^d , P{ (t) = a, for some t ∈ R _> ^N } = and the probability that (t) has k-multiple points is 1 or 0 according as whether 2kN > d(k − 1)β or 2kN < d(k−1)α. These results contain and extend the results of the Brownian sheet, where R _> ^N = (0,+∞) ^N ,R ₊ ^N =[10,+∞) ^N ,0< α≤1 and β≥1. Biography: LI Huiqiong (1966–), female, Associate professor, research direction: stochastic process and random fractal.     

Generalized Galerkin approximations for pseudoinverses and operator equations of the first kind

0 IntroductionLeqtuiliHnebaer afo rHimlbaenrtd sap baoceun.dLeedtlianaenardfufnbceti oan baolu onndeHd ,serse--spectively. Our goal is to seekv∈Hsuchthata(u,v) =f(u) , u∈ H (1)As a rule, we call the expression (1) a variational equationandva solutiontothis equation.By virtue of the Riesz repre-sentationtheorem,there exists a uniquely determined boundedlinear operatorA:H→Hsuchthata(u,v) =〈u,Av〉, u,v∈ H,where〈·,·〉is the inner product onH.If the sesquilinearformais also with posit…     

A class of complex quasi-Orlicz spaces and their analytic RN property

LetX be a complex quasi-Banach space and Φ:[0,∞)→[0,∞) an increasing convex function with Φ(0)=0, and Φ∈Δ₂. ThenL _Φ ^* (X) is a quasi-Banach space with continuous quasi-norm andL _Φ ^* (X) has the ARNP if and only ifX does. Supported by the National Natural Science Foundation of China Liu Peide: born in 1943, Professor     

The comparative investigation on redox property and second-order nonlinear response of Keggin-type α-[PM12O39NPh]3− (M = W and Mo) and Mo6NPh

Keggin-type phenylimido-polyoxometalates α-[PM₁₂O₃₉NPh]³⁻ (M = W and Mo) have been systematically investigated on the electronic structures, redox as well as nonlinear optical (NLO) properties by density functional theory (DFT). The strong M≡N bond confirmed by natural bond orbital (NBO) analysis comprises one s bond and two π bonds, the same as Mo≡N in [Mo₆O₁₈NPh]²⁻. Furthermore, phenylimido segment effectively modifies the electronic properties of α-[PM₁₂O₃₉NPh]³⁻. On one hand, when enlarging the inorganic cluster from {Mo₆O₁₈} to {PMo₁₂O₃₉}, the energy gap between HOMO and LUMO in α-[PMo₁₂O₃₉NPh]³⁻ decreased, resulting in enormously anodic shift for the reduction potential, while the excitation energy is less and the total second-order polarizability β ₀ is up to 438-3×10⁻³⁰ esu, which is nearly 10 times larger than that of [Mo₆O₁₈NPh]²⁻. On the other hand, when metal W in α-[PM₁₂O₃₉NPh]³⁻ is substituted by Mo, the interaction between Mo and N is enhanced and the redox ability becomes stronger. The β ₀ value for α-[PMo₁₂O₃₉NPh]³⁻ is more than 5 times higher than that of α-[PW₁₂O₃₉NPh]³⁻. It indicates that changing appropriate metal or enlarging the inorganic cluster will improve the redox properties and second-order nonlinear response. Moreover, the electron transition for three compounds mentioned above occurred mainly from organoimido segment (as the electron donor) to polyanion cluster (as the acceptor). As a result, α-[PMo₁₂O₃₉NPh]³⁻ may be a promising candidate for oxidant and nonlinear optical material. Electronic Supplementary Material  Supplementary material is available for this article at and is accessible for authorized users. Supported by National Natural Science Foundation of China (Grant No. 20573016), Training Fund of NENU’s Scientific Innovation Project (Grant No. NENUSTC07017), Science Foundation for Young Teachers of Northeast Normal University (Grant No. 20070304) and Program for Changjiang Scholars and Innovative Research Team in University (PCSIRT)     

A Large Deviation Principle for the Risk Process with Varying Premium

Let u ∈ R ,for any ω 〉 0, the processes X^ε = {X^ε（t）; 0 ≤ t≤ 1} are governed by the following random evolution equations dX^ε（t）= b（X^ε（t）,v（t））dt-εdSt/ε, where S={St; 0≤t≤1} is a compound Poisson process, the process v={v（t）; 0≤t≤1} is independent of S and takes values in R^m. We derive the large deviation principle for{（X^ε,v（.））; ε〉0} when ε↓0 by approximation method and contraction principle, which will be meaningful for us to find out the path property for the risk process of this type.     

Almost sure convergence and complete convergence for the weighted sums of martingale differences

Let {(D _n, FFF_n),n/->1} be a sequence of martingale differences and {a _ni, 1≤i≤n,n≥1} be an array of real constants. Almost sure convergence for the row sums are discussed. We also discuss complete convergence for the moving average processes underB-valued martingale differences assumption. Foundation item: Supported by the National Natural Science Foundation of China and the Doctoral Programme Foundation of China Biography: DENG Ai-jiao (1974-), female, Ph.D. candidate. Research interest is in stochastic processes and random fractal.