The Laws of Large Numbers Compared （大量的法律进行比较）.pdf

The Laws of Large Numbers Compared Tom Verhoeff July 1993 1 Introduction Probability Theory includes various theorems known as Laws of Large Numbers ; for instance, see [Fel68, Hea71, Ros89]. Usually two major categories are distin- guished: Weak Laws versus Strong Laws. Within these categories there are numer- ous subtle variants of differing generality. Also the Central Limit Theorems are often brought up in this context. Many introductory probability texts treat this topic superficially, and more than once their vague formulations are misleading or plainly wrong. In this note, we consider a special case to clarify the relationship between the Weak and Strong Laws. The reason for doing so is that I have not been able to find a concise formal exposition all in one place. The material presented here is certainly not new and was gleaned from many sources. In the following sections, X 1 X 2 is a sequence of independent and identi- cally distributed random variables withfinite expectation . We define the associ- ated sequence X i of partial sample means by n 1 X n X i n i1 The Laws of Large Numbers make statements about the convergence of X n to . Both laws relate bounds on sample size, accuracy of approximation, and degree of confidence. The Weak Laws deal with limits of probabilities involving X n . The Strong Laws deal with probabilities involving limits of X n . Especially the math- ematical underpinning of the Strong Laws requires a careful approach ([Hea71, Ch. 5] is an accessible presentation). 2 The Weak Law of Lar