The Laws of Large Numbers Compared
نویسنده
چکیده
Probability Theory includes various theorems known as Laws of Large Numbers; for instance, see [Fel68, Hea71, Ros89]. Usually two major categories are distinguished: Weak Laws versus Strong Laws. Within these categories there are numerous 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, X1, X2, . . . is a sequence of independent and identically distributed random variables with finite expectation μ. We define the associated sequence X̄i of partial sample means by X̄n = 1 n n ∑
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