文献类型：期刊文献

中文题名：深入剖析概率论与数理统计中的四种收敛定义及其反例:教学视角下的理解与辨析

英文题名：A Deep Analysis of the Convergence Definitions and Their Counterexamples in Probability and Mathematical Statistics:Understanding and Differentiation from a Teaching Perspective

作者：田苗[1]

机构：[1]华东理工大学数学学院,上海200237

年份：2026

卷号：42

期号：1

起止页码：77

中文期刊名：大学数学

外文期刊名：College Mathematics

基金：高等教育科学研究规划课题(25SX0313)。

语种：中文

中文关键词：收敛性;反例;知识理解

外文关键词：convergence;counterexample;knowledge understanding

摘要：随机变量序列的收敛性(依概率收敛、依分布收敛、r阶收敛和几乎处处收敛)因其高度抽象性,是学生学习过程中的难点.本文通过巧妙设计构造具体反例,系统揭示了收敛性之间的差异和联系.教学实践表明,将反例融入教学,学生对收敛概念的掌握程度显著提升,学习兴趣与主动性明显增强.
The convergence of sequences of random variables(convergence in probability,convergence in distribution,convergence in r-th mean,and almost sure convergence)is highly abstract and poses significant challenges for students.This paper designs specific counterexamples to systematically reveal the differences and connections between these types of convergence.Teaching practices have shown that incorporating counterexamples into instruction significantly enhances students'understanding of convergence concepts and markedly boosts their interest and initiative in learning.