Editing Law of large numbers (section)

== Limitation ==
The average of the results obtained from a large number of trials may fail to converge in some cases. For instance, the average of ''n'' results taken from the [[Cauchy distribution]] or some [[Pareto distribution]]s (α<1) will not converge as ''n'' becomes larger; the reason is [[Heavy-tailed distribution|heavy tails]].<ref>{{Cite book |title=A modern introduction to probability and statistics: understanding why and how |date=2005 |publisher=Springer |isbn=978-1-85233-896-1 |editor-last=Dekking |editor-first=Michel |series=Springer texts in statistics |location=London [Heidelberg] |pages=187}}</ref> The Cauchy distribution and the Pareto distribution represent two cases: the Cauchy distribution does not have an expectation,<ref>{{Cite book|title=A Modern Introduction to Probability and Statistics|url=https://archive.org/details/modernintroducti00fmde|url-status=dead| url-access=limited| last=Dekking|first=Michel|publisher=Springer|year=2005|isbn=9781852338961|pages=[https://archive.org/details/modernintroducti00fmde/page/n102 92]}}</ref> whereas the expectation of the Pareto distribution (''α''<1) is infinite.<ref>{{Cite book|title=A Modern Introduction to Probability and Statistics|url=https://archive.org/details/modernintroducti00fmde|url-status=dead|url-access=limited| last=Dekking|first=Michel| publisher=Springer| year=2005| isbn=9781852338961| pages=[https://archive.org/details/modernintroducti00fmde/page/n74 63]}}</ref> One way to generate the Cauchy-distributed example is where the random numbers equal the [[tangent]] of an angle uniformly distributed between −90° and +90°.<ref>{{Cite journal |last1=Pitman |first1=E. J. G. |last2=Williams |first2=E. J. |date=1967 |title=Cauchy-Distributed Functions of Cauchy Variates |journal=The Annals of Mathematical Statistics |volume=38 |issue=3 |pages=916–918 |doi=10.1214/aoms/1177698885 |jstor=2239008 |issn=0003-4851|doi-access=free }}</ref> The [[median]] is zero, but the expected value does not exist, and indeed the average of ''n'' such variables have the same distribution as one such variable. It does not converge in probability toward zero (or any other value) as ''n'' goes to infinity.

If the trials embed a [[selection bias]], typical in human economic/rational behaviour, the law of large numbers does not help in solving the bias, even if the number of trials is increased the selection bias remains.