Editing Law of large numbers (section)

===Differences between the weak law and the strong law===
The ''weak law'' states that for a specified large ''n'', the average <math style="vertical-align:-.35em">\overline{X}_n</math> is likely to be near ''μ''.<ref>{{Cite web |title=What Is the Law of Large Numbers? (Definition) {{!}} Built In |url=https://builtin.com/data-science/law-of-large-numbers |access-date=2023-10-20 |website=builtin.com |language=en}}</ref> Thus, it leaves open the possibility that <math style="vertical-align:-.4em">|\overline{X}_n -\mu| > \varepsilon</math> happens an infinite number of times, although at infrequent intervals. (Not necessarily <math style="vertical-align:-.4em">|\overline{X}_n -\mu| \neq 0</math> for all ''n'').

The ''strong law'' shows that this [[almost surely]] will not occur. It does not imply that with probability 1, we have that for any {{math|''ε'' > 0}} the inequality <math style="vertical-align:-.4em">|\overline{X}_n -\mu| < \varepsilon</math> holds for all large enough ''n'', since the convergence is not necessarily uniform on the set where it holds.<ref>{{harvtxt|Ross|2009}}</ref>

The strong law does not hold in the following cases, but the weak law does.<ref name="Weak law converges to constant">{{cite book |last1=Lehmann |first1=Erich L. |last2=Romano |first2=Joseph P. |date=2006-03-30 |title=Weak law converges to constant |publisher=Springer |isbn=9780387276052 |url=https://books.google.com/books?id=K6t5qn-SEp8C&pg=PA432}}</ref><ref>{{cite journal| title=A Note on the Weak Law of Large Numbers for Exchangeable Random Variables |author1=Dguvl Hun Hong |author2=Sung Ho Lee |url=http://www.mathnet.or.kr/mathnet/kms_tex/31810.pdf |journal=Communications of the Korean Mathematical Society| volume=13|year=1998|issue=2|pages=385–391 |access-date=2014-06-28|archive-url=https://web.archive.org/web/20160701234328/http://www.mathnet.or.kr/mathnet/kms_tex/31810.pdf|archive-date=2016-07-01|url-status=dead}}</ref><!-- Stack Exchange is not a reliable source -->

{{ordered list
|1= Let X be an [[Exponential distribution|exponentially]] distributed random variable with parameter 1. The random variable <math>\sin(X)e^X X^{-1}</math> has no expected value according to Lebesgue integration, but using conditional convergence and interpreting the integral as a [[Dirichlet integral]], which is an improper [[Riemann integral]], we can say:

<math display="block"> E\left(\frac{\sin(X)e^X}{X}\right) =\ \int_{x=0}^{\infty}\frac{\sin(x)e^x}{x}e^{-x}dx = \frac{\pi}{2} </math>

|2= Let X be a [[Geometric distribution|geometrically]] distributed random variable with probability 0.5. The random variable <math>2^X(-1)^X X^{-1}</math> does not have an expected value in the conventional sense because the infinite [[Series (mathematics)|series]] is not absolutely convergent, but using conditional convergence, we can say:

<math display="block"> E\left(\frac{2^X(-1)^X}{X}\right) =\ \sum_{x=1}^{\infty}\frac{2^x(-1)^x}{x}2^{-x}=-\ln(2) </math>

|3= If the [[cumulative distribution function]] of a random variable is

<math display="block">\begin{cases}
1-F(x)&=\frac{e}{2x\ln(x)},&x \ge e \\
F(x)&=\frac{e}{-2x\ln(-x)},&x \le -e
\end{cases}</math>

then it has no expected value, but the weak law is true.<ref>{{cite web|last1=Mukherjee|first1=Sayan|title=Law of large numbers| url=http://www.isds.duke.edu/courses/Fall09/sta205/lec/lln.pdf|access-date=2014-06-28|archive-url=https://web.archive.org/web/20130309032810/http://www.isds.duke.edu/courses/Fall09/sta205/lec/lln.pdf|archive-date=2013-03-09| url-status=dead}}</ref><ref>{{cite web|last1=J. Geyer|first1=Charles|title=Law of large numbers| url=http://www.stat.umn.edu/geyer/8112/notes/weaklaw.pdf}}</ref>

|4= Let ''X''<sub>''k''</sub> be plus or minus <math display="inline">\sqrt{k/\log\log\log k}</math> (starting at sufficiently large ''k'' so that the denominator is positive) with probability {{frac|1|2}} for each.<ref name=EMStrong/> The variance of ''X''<sub>''k''</sub> is then <math display="inline">k/\log\log\log k.</math> Kolmogorov's strong law does not apply because the partial sum in his criterion up to ''k''&nbsp;=&nbsp;''n'' is asymptotic to <math>\log n/\log\log\log n</math> and this is unbounded. If we replace the random variables with Gaussian variables having the same variances, namely <math display="inline">\sqrt{k/\log\log\log k}</math>, then the average at any point will also be normally distributed. The width of the distribution of the average will tend toward zero (standard deviation asymptotic to <math display="inline">1/\sqrt{2\log\log\log n}</math>), but for a given ''ε'', there is probability which does not go to zero with ''n'', while the average sometime after the ''n''th trial will come back up to ''ε''. Since the width of the distribution of the average is not zero, it must have a positive lower bound ''p''(''ε''), which means there is a probability of at least ''p''(''ε'') that the average will attain ε after ''n'' trials. It will happen with probability ''p''(''ε'')/2 before some ''m'' which depends on ''n''. But even after ''m'', there is still a probability of at least ''p''(''ε'') that it will happen. (This seems to indicate that ''p''(''ε'')=1 and the average will attain ε an infinite number of times.)
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