EXACT EVALUATIONS OF FINITE TRIGONOMETRIC SUMS BY SAMPLING THEOREMS

We use the sampling representations associated with Sturm-Liouville difference operators to derive generalized integral-valued trigonometric sums. This extends the known results where zeros of Chebyshev polynomials of the first kind are involved to the use of the eigenvalues of difference operators, which leads to new identities. In these identities Bernoulli＇s numbers play a role similar to that of Euler＇s in the old ones. Our technique differs from that of Byrne-Smith （1997） and Berndt-Yeap （2002）.

A THEOREM ON THE CONVERGENCE OF SUMS OF INDEPENDENT RANDOM VARIABLES

Let Sigma (infinity)(n=1) X-n be a series of independent random variables with at least one non-degenerate X-n, and let F-n be the distribution function of its partial sums S-n = Sigma (n)(k=1) X-k. Motivated by Hildebrand's work in [1], the authors investigate the a.s. convergence of Sigma (infinity)(n=1) X-n under a hypothesis that Sigma (infinity)(n=1) rho (X-n, c(n)) = infinity whener Sigma (infinity)(n=1) c(n) diverges, where the notation rho (X,c) denotes the Levy distance between the random variable X and the constant c. The principal result of this paper shows that the hypothesis is the condition under which the convergence of F-n(x(0)) with the limit value 0 < L-0 < 1, together with the essential convergence of Sigma (infinity)(n=1) X-n, is both sufficient and necessary in order for the series Sigma (infinity)(n=1) X-n to a.s. coverage. Moreover, if the essential convergence of Sigma (infinity)(n=1) X-n is strengthened to limsup(n=infinity) P(\S-n\ < K) = 1 for some K > 0, the hypothesis is already equivalent to the a.s. convergence of Sigma (infinity)(n=1) X-n. Here they have not only founded a very general limit theorem, but improved the related result in Hildebrand([1]) as well.

An Almost Sure Central Limit Theorem for Weighted Sums of Mixing Sequences

In this paper, we prove an almost sure central limit theorem for weighted sums of mixing sequences of random variables without stationary assumptions. We no longer restrict to logarithmic averages, but allow rather arbitrary weight sequences. This extends the earlier work on mixing random variables

A Note on the Almost Sure Central Limit Theorem for Partial Sums of ρ^(−)-Mixing Sequences

Let be a strictly stationary sequence of ρ?-mixing random variables. We proved the almost sure central limit theorem, containing the general weight sequences, for the partial sums , where , . The result generalizes and improves the previous results.

Almost sure central limit theory for products of sums of partial sums

Consider a sequence of i.i.d.positive random variables.An universal result in almost sure limit theorem for products of sums of partial sums is established.We will show that the almost sure limit theorem holds under a fairly general condition on the weight dk= k-1 exp（lnβk）,0≤β〈1.And in a sense,our results have reached the optimal form.

对于调和级数sum from n=1 to ∞(1/n)的分析

调和级数 ∑∞n =11n 是一种比较简单的发散级数 ,有关它发散性的证明 。

Almost Sure Central Limit Theorems for Heavily Trimmed Sums

We obtain an ahnost sure central limit theorem(ASCLT)for heavily trimmed sums.We also prove a function-typed ASCLT under the same conditions that assure measurable functions to satisfy the ASCLT for the partial sums of i.i.d,random variables with EX_1=0,EX_1~2=1.

The Almost Sure Central Limit Theorems for the Maxima of Sums under Some New Weak Dependence Assumptions

We prove the almost sure central limit theorems for the maxima of partial sums of r.v.＇s under a general condition of dependence due to Doukhan and Louhichi. We will separately consider the centered sequences and the sequences with positive expected values.

I.I.D.随机变量部分和之随机和的极限定理

论文研究了部分和之随机和的大数律和中心极限定理 ,所得结果推广了文献[4 ]中部分和之和的大数律和中心极限定理 .此外 ,论文还研究了由部分和之和所定义的停时 。

强混合序列部分和乘积的渐近正态性

设{Xn,n≥1}是同分布正的强混合随机变量序列.利用强混合序列的中心极限定理以及大数定律,在适当的条件下证明了∏nk=1Skn!μn1/(γσn)deN,n→∞,其中Sk=∑ki=1Xi,μ=EX1>0,σ2=VarX1<∞,γ=σμ,σn2=Var1γ∑kn=1kSμk-1,N为标准正态随机变量.

相依序列加权和的几乎处处中心极限定理

该文讨论了非平稳负(正)相依序列加权和的几乎处处中心极限定理,改进并推广了相依序列几乎处处中心极限定理的相关结果.

一类非线性分数阶微分方程边值问题正解的存在唯一性

运用和算子的不动点定理,研究了一类非线性分数阶微分方程边值问题正解的存在唯一性.结果不仅保证了正解的存在唯一性,而且能够构造一个迭代序列逼近它.最后,给出了一个例子说明所得结果的有效性.

多元Stancu算子的Boolean和迭代

定义单纯形上高维Stancu算子的Boolean和迭代算子并且研究它逼近连续函数的正定理、逆定理与饱和定理,得到了较高的逼近阶.

弱相依随机序列的若干新结果(英文)

设{Xn,n≥1}为-混合序列. 利用随机变量的截尾方法和-混合序列的三级数定理,讨论了-混合序列的收敛性质,并且得到了-混合序列的一类强极限定理,这些结果推广了独立序列的相应结果.最后研究了-混合序列加权和的强稳定性.

ρ^--混合序列部分和之和乘积的几乎处处中心极限定理

在适当的假设条件下,利用混合序列加权和的中心极限定理及矩不等式,证明了混合序列部分和之和乘积的几乎处处中心极限定理.

φ混合序列自正则加权和的中心极限定理

设{Xn,n≥1}为一严平稳φ混合随机变量序列,EX=0,V2n=∑ni=1Xi2,{an,i,1≤i≤n,n≥1}为一实数阵列,Sn=∑ni=1an,iXi.利用随机变量阵列的弱收敛定理,在较一般的条件下,证明了自正则加权和{Sn/Vn,n≥1}的中心极限定理,改进并推广了已有混合序列自正则化中心极限定理的相关结果.

关于和式极限的两个定理及其应用

从2017年3月第八届全国大学生数学竞赛决赛(数学类)的一道题入手,介绍了两个相关和式极限的定理,推广了一些结果.这些结论有助于完善相关教学内容.

部分和乘积的几乎处处中心极限定理

设{X_n,n≥1}是独立同分布正的随机变量序列,E(X_1)=μ>0,Var(X_1)=σ~2,变异系数γ=σ/μ.g是满足一定条件的无界可测函数,证明了其中F(·)是随机变量e^(2^(1/2)ξ)的分布函数,ξ是服从标准正态分布的随机变量.

关于部分和之和乘积的注记

设{X,Xn,n≥1}是独立同分布的正的均方可积的随机变量序列,μ=E(X)>0,σ2=Var(X),变异系数γ=σ/μ,记Tn=∑nk=1kXk,得到了∏Nn=1Tn在某种正则化因子下的极限分布.

φ-混合序列部分和乘积的几乎处处中心极限定理

关于一列独立同分布正随机变量部分和乘积的几乎处处中心极限定理,已得出了结果.本文把独立性推广到相依随机变量的情形,在φ-混合序列部分和乘积的渐近对数正态性基础上,以一个三角列的几乎处处中心极限定理为跳板,证明了在∑from n=1 to ∞φ(1/2)(n)<∞,且0<σ02=1+2∑ from j=1 ∞ E〔(X1-μ)/σ〕〔(Xj+1-μ)/σ〕<∞的条件下的几乎处处中心极限定理.