If the Strong Law of Large Numbers imply the Weak Law, why do we have a Weak Law of Large Numbers?


closed as unclear what you're asking by Namaste, Shailesh, Kavi Rama Murthy, user593746, BigbearZzz Dec 15 '18 at 17:18

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    $\begingroup$ Because the weak law does not imply the strong law. $\endgroup$ – drhab Dec 15 '18 at 12:14
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    $\begingroup$ And because we have several types of convergence in probability theory $\endgroup$ – Fakemistake Dec 15 '18 at 12:54

A weak law may come with some estimate of the rate of convergence which is not available for the strong law. In practical situations, such as statistics, you never have a whole infinite sequence of trials, only a finite sequence. Yet you would still like to deduce something about the underlying probability distribution.

The textbook

Chung, Kai Lai, Elementary probability theory with stochastic processes, Undergraduate Texts in Mathematics. New York - Heidelberg - Berlin: Springer-Verlag. X, 325 p. Cloth DM 29.40; $ 12.00 (1974). ZBL0293.60001.

perplexes students by showing two quotes on the same page, one saying that the strong law is superior to the weak, and the other saying the weak law is superior to the strong. This leaves the poor instructor (me) to explain the discrepancy!

Here it is, from page 233 in the first edition:

Feller: "[the weak law of large numbers] is of very limited interest and hould be replaced by the more precise and more useful strong law of large numbers" (p. 152 of An Introduction to Probability Theory and its Applications, vol I, 3rd edition, 1971).

van der Waerden: "[the strong law of large numbers] scarcely plays a role in mathematical statistics" (p. 98 of Mathematische Statistik, 3rd ed, 1971)


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