Question: Are there two different Weak Laws of Large Numbers in the statistics literature?

I have found many sources which claim that the weak law of large numbers requires existence of finite second moments, while the result I know doesn't even require the existence of first moments.

In particular, I am debating returning a mathematical statistics textbook I purchased, because, if there aren't two different results in the literature under the name "Weak Law of Large Numbers", then the authors seem to have made a major mistake regarding introductory material. This decreases my confidence substantially about the accuracy of the rest of the book.

Background: On p. 204 of Craig, Hogg, Introduction to Mathematical Statistics, it says:

... In a more advanced course a Strong Law of Large Numbers is proven... One result of this theorem [the Strong Law of Large Numbers] is that we can weaken the hypothesis of Theorem 4.2.1. [the Weak Law of Large Numbers] to the assumption that the random variables $X_i$ are independent and each has finite mean $\mu$. Thus the Strong Law of Large Numbers is a first moment theorem, while the Weak Law requires the existence of a second moment.

Also, the page for the Weak Law of Large Numbers from Wolfram MathWorld also claims that it requires existence of the second moment, as does the textbook quoted in this question on CrossValidated. This webpage also uses a proof assuming finite variance and mean square convergence (which of course implies convergence in probability). A similar result, under the name $L^2$ weak law is found on p.55, Theorem 2.2.3 of Durrett Probability: Theory and Examples. These results are weaker than that of the strong law, and are proved under less general conditions.

However, for the Weak Law of Large Numbers that one finds e.g. on Wikipedia or Durrett Probability: Theory and Examples, p. 60, Theorem 2.2.7., much weaker assumptions than the existence of the second moment, in fact, even strictly weaker than existence of the first moment, are required (see also this answer on CrossValidated):

Let $X_1, X_2, \dots,$ be i.i.d. with $$x \mathbb{P}(|X_i|>x) \to 0 \quad as \quad x \to \infty $$ Let $S_n = X_1 + \dots + X_n$ and let $\mu_n = \mathbb{E}[X_1 1_{|X_1| \le n}]$. Then $S_n/n - \mu_n \to 0$ in probability.

As one can see, the conclusion of this theorem is weaker than the Strong Law of Large numbers, but its assumptions are also more general.

This is, of course, what one would expect -- theorems with weaker conclusions should usually be provable under conditions more general than theorems with stronger conclusions -- otherwise, the theorem with the weaker conclusions would just be a special case of the theorem with the stronger conclusions and not a result in its own right. (Again, see this question on CrossValidated.)

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    $\begingroup$ can you include the statement of theorem 4.2.1 in Hogg and Craig? $\endgroup$ – user795305 Jun 30 '17 at 17:21
  • $\begingroup$ @Ben "(Weak Law of Large Numbers) Let $\{ X_n \}$ be a sequence of i.i.d. random variables having common mean $\mu$ and variance $\sigma^2 < \infty$. Let $\bar{X}_n = n^{-1} \sum_{i=1}^n X_i$. Then $$\bar{X}_n \overset{P}{\to} \mu$$" $\endgroup$ – Chill2Macht Jun 30 '17 at 17:31
  • $\begingroup$ Is $\stackrel{P}{\to}$ defined in the usual sense as convergence in probability? I was hoping that it was just a version of thm 2.2.3 of durrett $\endgroup$ – user795305 Jun 30 '17 at 17:33
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    $\begingroup$ @Ben Yes, I believe it does mean convergence in probability. $\endgroup$ – Chill2Macht Jun 30 '17 at 17:39
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    $\begingroup$ As a general note: the version of the WLLN I quote (the one I am familiar with) is apparently called the Feller WLLN or the Kolmogorov-Feller WLLN, see here: www2.stat.duke.edu/courses/Fall09/sta205/lec/lln.pdf or here link.springer.com/article/10.1023/B:JOTP.0000040299.15416.0c Also, apparently conditions for the Kolmogorov-Feller WLLN are not only sufficient but also necessary; see this document: stat.umn.edu/geyer/8112/notes/weaklaw.pdf $\endgroup$ – Chill2Macht Jun 30 '17 at 17:40

Many results in statistics have generic names that apply to a collection of theorems asserting some result under different conditions. My understanding is that a "weak law of large numbers" can refer to any theorem that shows convergence-in-probability of the sequence of sample means to a corresponding mean. Any theorem that specifies sufficient conditions for this kind of convergence in the context of some model could legitimately be regarded as a "weak law of large numbers". Hence, there could potentially be hundreds of specific theorems under the general rubric of the "weak law of large numbers", each giving conditions for convergence in some context.

(Similarly, reference to the "strong law of large numbers" can refer to any theorem showing almost-sure convergence of a sequence of means to a corresponding mean parameter, and reference to a "central limit theorem" can refer to any theorem that asserts distributional convergence of a scaled sequence of sample means. All these terms can be used generically to refer to a class of theorems or specifically to refer to particular theorems.)


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