关于一些概率统计理论的一点认识

近来,林群从哲学的角度用公式相对真理/绝对真理=0.9阐述了微积分的内涵,并探究了计算数学、概率统计等学科领域乃至日常生活现象也同样蕴藏着该哲学公式.基于林群等在《概率论初步设想》中对大数定律和中心极限定理等概率统计学理论的哲学内涵分析,用一些数值算例展示概率统计学中这些常用理论所蕴含的这一哲学公式.     

以学生为中心的案例讨论式教学——大数定律

探讨工科概率统计课程中大数定律的一种不同常规的教学设计及实施过程.此设计充分体现了以学生为教学过程主体的原则,通过学生感兴趣的实际案例进行问题引入,按特殊到一般的顺序引导学生对几个常见大数定律由浅至深进行讨论式学习.同时辅以课堂实验、计算机模拟、数学建模等手段帮助学生深入理解大数定律结论及条件的含义,掌握其工程应用背景,并总结了大数定律的课程思政内容.     

Peng g- 期望下的大数定律

Peng 于1997 年通过倒向随机微分方程引入了一类性质很好的非线性数学期望, 即g- 期望. 本文中, 我们将给出Peng g- 期望下的弱大数定律与强大数定律.     

ρ混合序列的大数定律和完全收敛性

主要研究了ρ混合序列的大数定律和完全收敛性,获得了与独立情形一样的大数定律和完全收敛定理.     

φ混合序列的大数定律和完全收敛性

主要研究了φ混合序列的大数定律和完全收敛性,获得了与独立情形一样的大数定律和完全收敛定理.     

■混合序列的大数定律和完全收敛性

主要研究了■混合序列的大数定律和完全收敛性,获得了与独立情形一样的大数定律和完全收敛定理.     

两两NQD列的大数定律和完全收敛性

本文主要研究了两两NQD列的的大数定律和完全收敛性,获得了与独立情形一样的大数定律和完全收敛定理.     

树指标m重非齐次马氏信源的一个强大数定律

树指标随机过程已成为近年来发展起来的概率论的研究方向之一.在概率论的发展过程中,对强大数定律的研究一直占重要地位,强大数定律也一直是国际概率论界研究的中心课题之一.将经典的一重非齐次马氏信源上的强大数定律推广到树指标m重非齐次马氏信源上,以拓展强大数定律的适用范围.通过引入相对熵密度偏差的概念和构造非负鞅,将Doob鞅收敛定理应用于几乎处处收敛的研究,研究给出了关于树上m重非齐次马氏信源的一个强大数定律.     

次线性期望下大数定律的收敛速率（英文）

在本文中,我们研究次线性期望下独立同分布随机变量的大数定律的收敛速率.我们给出了大数定律的一个强L~p收敛版本和一个强拟必然收敛版本.     

(～ρ)混合序列的大数定律和完全收敛性

本文研究了(～ρ)混合序列的大数定律和完全收敛性.利用Bryc W.和Smolenski W.不等式,获得了与独立情形一样的大数定律和完全收敛定理.     

Uniform convergence of reversed martingales

A necessary and sufficient condition for the uniform convergence of a family of reversed martingales converging to a degenerated limiting process is given. The condition is expressed by means of regular convergence (in Hardy's sense) of corresponding means. It is shown that the given regular convergence is equivalent to Hoffmann-Jørgensen's eventually totally boundedness in the mean which is necessary and sufficient for the uniform law of large numbers. Analogous results are carried out for families of reversed submartingales. By applying derived results several convergence statements are obtained which extend those from the uniform law of large numbers to the general reversed martingale case.     

两两NQD阵列行和的若干极限定理

讨论了两两NQD阵列行和的弱收敛性、L_p收敛性和完全收敛性,在{X_(nk);1≤k≤k_n↑∞,n≥1}是Cesaro一致可积的相关条件下,获得了两两NQD阵列行和的弱收敛性、Lp收敛性和完全收敛性定理,将独立阵列行和的相关极限定理推广到了两两NQD阵列行和的情形.     

The Convergence of a Moving Average Process of AANA Random Variables

Based on the asymptotically almost negatively associated(AANA) random vari-ables, we investigate the complete moment convergence for a moving average process under the moment condition E[Y log(1+Y )] < ∞. As an application, Marcinkiewicz-Zygmund-type strong law of large numbers for this moving average process is presented in this paper.     

The strong law of large numbers for random processes

Analogs of the Kolmogorov, Zygmund-Martsenkevich, and Brunk-Prokhorov strong law of large numbers are proved for martingales with continuous parameter. A new generalization of the Brunk—Prokhorov strong law of large numbers is given for martingales with discrete times. Along with convergence almost everywhere, we also prove the average convergence.     

Convergence Rates in the Law of Large Numbers under Sublinear Expectations

In this note, we study convergence rates in the law of large numbers for independent and identically distributed random variables under sublinear expectations. We obtain a strong L^p-convergence version and a strongly quasi sure convergence version of the law of large numbers.     

任意随机序列的变换及其a.s.收敛性

本文利用截尾方法构造几乎处处收敛的鞅结合无穷乘积定理，研究随机变量序列变换的局部收敛性及强大数定理，作为推论得到了关于赌博系统的若干强极限定理。     

On the rate of convergence in the strong law of large numbers for martingales

The aim of this note is to establish the Baum–Katz type rate of convergence in the Marcinkiewicz–Zygmund strong law of large numbers for martingales, which improves the recent works of Stoica [Series of moderate deviation probabilities for martingales, J. Math. Anal. Appl. 336 (2005), pp. 759–763; Baum–Katz–Nagaev type results for martingales, J. Math. Anal. Appl. 336 (2007), pp. 1489–1492; A note on the rate of convergence in the strong law of large numbers for martingales, J. Math. Anal. Appl. 381 (2011), pp. 910–913]. Furthermore, we also study some relevant limit behaviours for the uniform mixing process. Under some uniform mixing conditions, the sufficient and necessary condition of the convergence of the martingale series is established.     

Complete moment convergence of pairwise NQD random variables

It is known that the dependence structure of pairwise negative quadrant dependent (NQD) random variables is weaker than those of negatively associated random variables and negatively orthant dependent random variables. In this article, we investigate the moving average process which is based on the pairwise NQD random variables. The complete moment convergence and the integrability of the supremum are presented for this moving average process. The results imply complete convergence and the Marcinkiewicz–Zygmund-type strong law of large numbers for pairwise NQD sequences.