臺灣博碩士論文加值系統

本論文研究證明強大數法則成立的一些技巧和相關 的數學工具,討論如何
處理部分和序列的 幾乎到處(almost sure) 收斂 問題.即處理強大數法
則(S.L.L.N.)的問題.第一章探討文獻上證明強大數法則成立之各種基本
證明方法.為了解如何應用這些基本證明方法,我們用這些基本方法證明
Borel強大數法則 .希望於探討證明Borel強大數法則的過程中,能夠很快
的掌握證明大數法則的各種方法,及所需的數學工具.我們整理歸納出4種
證明強大數法則的基本方法.第1方法:隨機變數部分和算數平均數序列的
完備(complete)收斂,保證隨機變數部分和算數平均數序列的幾乎到處(a.
s.)收斂.第2方法 :利用隨機變數序列的子序列,創造隨機數部分和算數平
均數序列的上下界.第3方法:證明由隨機變數的算數平均數序列所構成的
無窮級數是收斂,根據 Kronecker 引理,知強大數法則成立.第4方法:證
明當n往無窮大跑時,n 項隨機變數部分和與其期望值差距的算術平均數,
大於任意正數的事件,會發生無窮多次的機率是零.並證明當n往無窮大跑
時,n項隨機變數部分和與其期望值差距的算術平均數,小於任意負數的事
件,會發生無窮多次的機率是零 .綜合上面兩個結果,得證強大數法則.第
二章對使用上述4個基本方法所需的數學工具,做廣泛的研究.用第 1基本
方法證明大數法則,需知到大數法則的收斂速率,我們探討部分和機率不等
式.第2基本方法的重要步驟是控制收斂子序列與原序列差距的收斂性.第3
基本方法與無窮級數的收斂有關,故無窮級數的收斂工具及結果,都納入討
論.用第4基本方法證明大數法則,必需探討動差母函數之上界,此與高斯
( Gaussian ) 隨 機 變 數 有 關 係 , 故我們探討高斯隨機變數之應
用.第三章探討大數法則成立之充分條件.第四章探討大數法則成立之必要
條件.

The techniques and the relative mathematical devices for
proving the strong law of large numbers (S.L.L.N.) are studied
in the thesis. We discuss how to slve the almost sure (a.s.)
convergence problems of the sequence of partial sum. That is to
deal with the S.L.L.N. The fundamental methods for proving the
S.L.L.N. are surveyed in chapter 1. For understanding the
application of the fundamental methods, the fundamental methods
are used to prove Borel's S.L.L.N.By virtue of this study
process, we will have a better and widespread understanding of
the allied mathematical devicse for proving S.L.L.N. 4
fundamental methods for proving S.L.N.N. are collected and
discussed. The extensive explorations on the mathematical
devices of the above 4 fundamental methods are proposed in
chapter 2. For better understanding method 1, we investigate
the probability inequalities of partial sum. The curcial step
in method 2 is to control the convergence of difference between
convergent subsequence and original sequence. Method 3 has
relation to the convergence of infinite series. The upper
bounds of moment generating functions are explored when
fundamental method 4 is used to prove S.S.L.N. the foregoing
upper bounds have relation to the Gaussian random variable, so
we study the application of the Gaussian random variable. The
sufficent and necessary conditions of S.L.L.N. are surveyed in
chapter 3 and 4 respectively.