Editing Law of large numbers

{{Distinguish|Law of truly large numbers}}

{{Short description|Averages of repeated trials converge to the expected value}}

{{Probability fundamentals}}

[[File:Lawoflargenumbers.svg|An [[illustration]] of the law of large numbers using a particular run of rolls of a single [[Dice|die]].  As the number of rolls in this run increases, the average of the values of all the results approaches 3.5.  Although each run would show a distinctive shape over a small number of throws (at the left), over a large number of rolls (to the right) the shapes would be extremely similar.|thumb|right|286x286px]]

In [[probability theory]], the '''law of large numbers''' is a [[Law (mathematics)|mathematical law]] that states that the [[average]] of the results obtained from a large number of independent random samples converges to the true value, if it exists.<ref name=":0">{{Cite book|title=A Modern Introduction to Probability and Statistics| url=https://archive.org/details/modernintroducti00fmde|url-access=limited| last=Dekking|first=Michel| publisher=Springer| year=2005|isbn=9781852338961|pages=[https://archive.org/details/modernintroducti00fmde/page/n191 181]–190}}</ref> More formally, the law of large numbers states that given a sample of independent and identically distributed values, the [[Sample mean and covariance|sample mean]] converges to the true [[mean]]. 

The law of large numbers is important because it guarantees stable long-term results for the averages of some [[Randomness|random]] [[Event (probability theory)|events]].<ref name=":0" /><ref>{{Cite journal|last1=Yao|first1=Kai|last2=Gao|first2=Jinwu|date=2016|title=Law of Large Numbers for Uncertain Random Variables|journal=IEEE Transactions on Fuzzy Systems| volume=24| issue=3| pages=615–621| doi=10.1109/TFUZZ.2015.2466080| s2cid=2238905|issn=1063-6706}}</ref> For example, while a [[casino]] may lose [[money]] in a single spin of the [[roulette]] wheel, its earnings will tend towards a predictable percentage over a large number of spins. Any winning streak by a player will eventually be overcome by the parameters of the game. Importantly, the law applies (as the name indicates) only when a ''large number'' of observations are considered. There is no principle that a small number of observations will coincide with the expected value or that a streak of one value will immediately be "balanced" by the others (see the [[gambler's fallacy]]).

The law of large numbers only applies to the ''average'' of the results obtained from repeated trials and claims that this average converges to the expected value; it does not claim that the ''sum'' of ''n'' results gets close to the expected value times ''n'' as ''n'' increases.

Throughout its history, many mathematicians have refined this law. Today, the law of large numbers is used in many fields including statistics, probability theory, economics, and insurance.<ref name=":1">{{Cite web |last=Sedor |first=Kelly |title=The Law of Large Numbers and its Applications |url=https://www.lakeheadu.ca/sites/default/files/uploads/77/images/Sedor%20Kelly.pdf}}</ref>

==Examples==
For example, a single roll of a six-sided [[dice]] produces one of the numbers 1, 2, 3, 4, 5, or 6, each with equal [[probability]].  Therefore, the [[expected value]] of the roll is:

<math display="block"> \frac{1+2+3+4+5+6}{6} = 3.5</math>

According to the law of large numbers, if a large number of six-sided dice are rolled, the average of their values (sometimes called the [[sample mean]]) will approach 3.5, with the precision increasing as more dice are rolled.

It follows from the law of large numbers that the [[empirical probability]] of success in a series of [[Bernoulli trial]]s will converge to the theoretical probability. For a [[Bernoulli random variable]], the expected value is the theoretical probability of success, and the average of ''n'' such variables (assuming they are [[Independent and identically distributed random variables|independent and identically distributed (i.i.d.)]]) is precisely the relative frequency.
[[File:Law_of_large_numbers_(black_%26_red_balls).png|thumb|295x295px| This image illustrates the convergence of relative frequencies to their theoretical probabilities. The probability of picking a red ball from a sack is 0.4 and black ball is 0.6. The left plot shows the relative frequency of picking a black ball, and the right plot shows the relative frequency of picking a red ball, both over 10,000 trials. As the number of trials increases, the relative frequencies approach their respective theoretical probabilities, demonstrating the law of large numbers.]]
For example, a [[fair coin]] toss is a Bernoulli trial. When a fair coin is flipped once, the theoretical probability that the outcome will be heads is equal to {{frac|1|2}}. Therefore, according to the law of large numbers, the proportion of heads in a "large" number of coin flips "should be" roughly {{frac|1|2}}. In particular, the proportion of heads after ''n'' flips will [[almost surely]] [[limit of a sequence|converge]] to {{frac|1|2}} as ''n'' approaches infinity.

Although the proportion of heads (and tails) approaches {{frac|1|2}}, almost surely the [[absolute difference]] in the number of heads and tails will become large as the number of flips becomes large. That is, the probability that the absolute difference is a small number approaches zero as the number of flips becomes large. Also, almost surely the ratio of the absolute difference to the number of flips will approach zero. Intuitively, the expected difference grows, but at a slower rate than the number of flips.

Another good example of the law of large numbers is the [[Monte Carlo method]]. These methods are a broad class of [[computation]]al [[algorithm]]s that rely on repeated [[random sampling]] to obtain numerical results. The larger the number of repetitions, the better the approximation tends to be. The reason that this method is important is mainly that, sometimes, it is difficult or impossible to use other approaches.<ref>{{Cite journal|last1=Kroese|first1=Dirk P.| last2=Brereton|first2=Tim| last3=Taimre|first3=Thomas|last4=Botev|first4=Zdravko I.|date=2014|title=Why the Monte Carlo method is so important today|journal=Wiley Interdisciplinary Reviews: Computational Statistics| language=en| volume=6| issue=6|pages=386–392|doi=10.1002/wics.1314|s2cid=18521840}}</ref>

== 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.

==History==
[[File:DiffusionMicroMacro.gif|thumb|right|upright=1.15|[[Molecular diffusion|Diffusion]] is an example of the law of large numbers. Initially, there are [[solute]] molecules on the left side of a barrier (magenta line) and none on the right. The barrier is removed, and the solute diffuses to fill the whole container.{{ubl|style=margin-top:1em|
''Top:'' With a single molecule, the motion appears to be quite random.
|''Middle:'' With more molecules, there is clearly a trend where the solute fills the container more and more uniformly, but there are also random fluctuations.
|''Bottom:'' With an enormous number of solute molecules (too many to see), the randomness is essentially gone: The solute appears to move smoothly and systematically from high-concentration areas to low-concentration areas. In realistic situations, chemists can describe diffusion as a deterministic macroscopic phenomenon (see [[Fick's law]]s), despite its underlying random nature.}}]]

The Italian mathematician [[Gerolamo Cardano]] (1501–1576) stated without proof that the accuracies of empirical statistics tend to improve with the number of trials.<ref>{{cite book |last=Mlodinow |first=L. |title=The Drunkard's Walk |location=New York |publisher=Random House |year=2008 |page=50}}</ref><ref name=":1" /> This was then formalized as a law of large numbers. A special form of the law of large numbers (for a binary random variable) was first proved by [[Jacob Bernoulli]].<ref>{{cite book |first=Jakob |last=Bernoulli |title=Ars Conjectandi: Usum & Applicationem Praecedentis Doctrinae in Civilibus, Moralibus & Oeconomicis |language=la |year=1713 |chapter=4 |translator-first=Oscar |translator-last=Sheynin}}</ref><ref name=":1" /> It took him over 20 years to develop a sufficiently rigorous mathematical proof which was published in his {{lang|la|italic=yes|[[Ars Conjectandi]]}} (''The Art of Conjecturing'') in 1713. He named this his "golden theorem" but it became generally known as "'''Bernoulli's theorem'''". This should not be confused with [[Bernoulli's principle]], named after Jacob Bernoulli's nephew [[Daniel Bernoulli]]. In 1837, [[Siméon Denis Poisson|S. D. Poisson]] further described it under the name {{lang|fr|"la loi des grands nombres"}} ("the law of large numbers").<ref>Poisson names the "law of large numbers" ({{lang|fr|la loi des grands nombres}}) in: {{cite book |first=S. D. |last=Poisson |title=Probabilité des jugements en matière criminelle et en matière civile, précédées des règles générales du calcul des probabilitiés |location=Paris, France |publisher=Bachelier |year=1837 |page=[https://archive.org/details/recherchessurla02poisgoog/page/n30 7] |language=fr}} He attempts a two-part proof of the law on pp. 139–143 and pp. 277 ff.</ref><ref>{{cite journal |last=Hacking |first=Ian |year=1983 |title=19th-century Cracks in the Concept of Determinism |journal=Journal of the History of Ideas |volume=44 |issue=3 |pages=455–475 |doi=10.2307/2709176 |jstor=2709176}}</ref><ref name=":1" /> Thereafter, it was known under both names, but the "law of large numbers" is most frequently used.

After Bernoulli and Poisson published their efforts, other mathematicians also contributed to refinement of the law, including [[Pafnuty Chebyshev|Chebyshev]],<ref>{{Cite journal | last1 = Tchebichef | first1 = P. | title = Démonstration élémentaire d'une proposition générale de la théorie des probabilités | doi = 10.1515/crll.1846.33.259 | journal = Journal für die reine und angewandte Mathematik | volume = 1846 | issue = 33 | pages = 259–267 | year = 1846 | s2cid = 120850863 | url = https://zenodo.org/record/1448850 |language=fr}}</ref> [[Andrey Markov|Markov]], [[Émile Borel|Borel]], [[Francesco Paolo Cantelli|Cantelli]], [[Andrey Kolmogorov|Kolmogorov]] and [[Aleksandr Khinchin|Khinchin]].<ref name=":1" /> Markov showed that the law can apply to a random variable that does not have a finite variance under some other weaker assumption, and Khinchin showed in 1929 that if the series consists of independent identically distributed random variables, it suffices that the [[expected value]] exists for the weak law of large numbers to be true.{{sfn|Seneta|2013}}<ref name=EncMath>{{cite web| author1=Yuri Prohorov|author-link1=Yuri Vasilyevich Prokhorov|title=Law of large numbers| url=https://www.encyclopediaofmath.org/index.php/Law_of_large_numbers| website=Encyclopedia of Mathematics |publisher=EMS Press}}</ref> These further studies have given rise to two prominent forms of the law of large numbers. One is called the "weak" law and the other the "strong" law, in reference to two different modes of [[limit of a sequence|convergence]] of the cumulative sample means to the expected value; in particular, as explained below, the strong form implies the weak.{{sfn|Seneta|2013}}

==Forms==
There are two different versions of the law of large numbers that are described below. They are called the'' '''strong law''' of large numbers'' and the '''''weak law''' of large numbers''.<ref>{{Cite book|title=A Course in Mathematical Statistics and Large Sample Theory| last1=Bhattacharya|first1=Rabi| last2=Lin|first2=Lizhen| last3=Patrangenaru|first3=Victor| date=2016| publisher=Springer New York| isbn=978-1-4939-4030-1| series=Springer Texts in Statistics| location=New York, NY| doi=10.1007/978-1-4939-4032-5}}</ref><ref name=":0" /> Stated for the case where ''X''<sub>1</sub>, ''X''<sub>2</sub>, ... is an infinite sequence of [[Independent and identically distributed random variables|independent and identically distributed (i.i.d.)]] [[Lebesgue integration|Lebesgue integrable]] random variables with expected value E(''X''<sub>1</sub>) = E(''X''<sub>2</sub>) = ... = ''μ'', both versions of the law state that the sample average

<math display="block">\overline{X}_n=\frac1n(X_1+\cdots+X_n) </math>

converges to the expected value:
{{NumBlk||<math display="block">\overline{X}_n \to \mu \quad\textrm{as}\ n \to \infty.</math>|{{EquationRef|1}}}}

(Lebesgue integrability of ''X<sub>j</sub>'' means that the expected value E(''X<sub>j</sub>'') exists according to Lebesgue integration and is finite. It does ''not'' mean that the associated probability measure is [[absolutely continuous]] with respect to [[Lebesgue measure]].)

Introductory probability texts often additionally assume identical finite [[variance]] <math> \operatorname{Var} (X_i) = \sigma^2 </math> (for all <math>i</math>) and no correlation between random variables.  In that case, the variance of the average of ''n'' random variables is

<math display="block">\operatorname{Var}(\overline{X}_n) = \operatorname{Var}(\tfrac1n(X_1+\cdots+X_n)) = \frac{1}{n^2} \operatorname{Var}(X_1+\cdots+X_n) = \frac{n\sigma^2}{n^2} = \frac{\sigma^2}{n}.</math>

which can be used to shorten and simplify the proofs. This assumption of finite [[variance]] is ''not necessary''. Large or infinite variance will make the convergence slower, but the law of large numbers holds anyway.<ref name="TaoBlog">{{cite web|title=The strong law of large numbers – What's new|date=19 June 2008|url=http://terrytao.wordpress.com/2008/06/18/the-strong-law-of-large-numbers/|access-date=2012-06-09|publisher=Terrytao.wordpress.com}}</ref> 

[[Independence (probability theory)#More than two random variables|Mutual independence]] of the random variables can be replaced by [[pairwise independence]]<ref>{{cite journal|last1=Etemadi|first1=N. Z.|date=1981|title=An elementary proof of the strong law of large numbers|journal=Wahrscheinlichkeitstheorie Verw Gebiete| volume=55| issue=1| pages=119–122| doi=10.1007/BF01013465|s2cid=122166046|doi-access=free}}</ref> or [[Exchangeable random variables|exchangeability]]<ref>{{Cite journal| last=Kingman|first=J. F. C.|date=April 1978|title=Uses of Exchangeability|journal=The Annals of Probability| language=en| volume=6|issue=2|doi=10.1214/aop/1176995566|issn=0091-1798|doi-access=free}}</ref> in both versions of the law.

The difference between the strong and the weak version is concerned with the mode of convergence being asserted. For interpretation of these modes, see [[Convergence of random variables]].

===Weak law===
{{multiple image |width1=50 |image1=Blank300.png
|width2=100 |image2=Lawoflargenumbersanimation2.gif |footer=Simulation illustrating the law of large numbers. Each frame, a coin that is red on one side and blue on the other is flipped, and a dot is added in the corresponding column. A pie chart shows the proportion of red and blue so far. Notice that while the proportion varies significantly at first, it approaches 50% as the number of trials increases.
|width3=50 |image3=Blank300.png}}
The '''weak law of large numbers''' (also called [[Aleksandr Khinchin|Khinchin]]'s law) states that given a collection of [[Independent and identically distributed random variables|independent and identically distributed]] (iid) samples from a random variable with finite mean, the sample mean [[Convergence in probability|converges in probability]] to the expected value<ref>{{harvnb|Loève|1977|loc=Chapter 1.4, p. 14}}</ref>
{{NumBlk||<math display="block">
    \overline{X}_n\ \overset{P}{\rightarrow}\ \mu \qquad\textrm{when}\ n \to \infty.
</math>|{{EquationRef|2}}}}

That is, for any positive number ''ε'',

<math display="block">
    \lim_{n\to\infty}\Pr\!\left(\,|\overline{X}_n-\mu| < \varepsilon\,\right) = 1.
</math>

Interpreting this result, the weak law states that for any nonzero margin specified (''ε''), no matter how small, with a sufficiently large sample there will be a very high probability that the average of the observations will be close to the expected value; that is, within the margin.

As mentioned earlier, the weak law applies in the case of i.i.d. random variables, but it also applies in some other cases. For example, the variance may be different for each random variable in the series, keeping the expected value constant. If the variances are bounded, then the law applies, as shown by [[Pafnuty Chebyshev|Chebyshev]] as early as 1867. (If the expected values change during the series, then we can simply apply the law to the average deviation from the respective expected values. The law then states that this converges in probability to zero.) In fact, Chebyshev's proof works so long as the variance of the average of the first ''n'' values goes to zero as ''n'' goes to infinity.<ref name=EncMath/> As an example, assume that each random variable in the series follows a [[Gaussian distribution]] (normal distribution) with mean zero, but with variance equal to <math>2n/\log(n+1)</math>, which is not bounded. At each stage, the average will be normally distributed (as the average of a set of normally distributed variables). The variance of the sum is equal to the sum of the variances, which is [[asymptotic]] to <math>n^2 / \log n</math>. The variance of the average is therefore asymptotic to <math>1 / \log n</math> and goes to zero.

There are also examples of the weak law applying even though the expected value does not exist.

===Strong law===
The '''strong law of large numbers''' (also called [[Andrey Kolmogorov|Kolmogorov]]'s law) states that the sample average [[Almost sure convergence|converges almost surely]] to the expected value<ref>{{harvnb|Loève|1977|loc=Chapter 17.3, p. 251}}</ref>
{{NumBlk||<math display="block">
    \overline{X}_n\ \overset{\text{a.s.}}{\longrightarrow}\ \mu \qquad\textrm{when}\ n \to \infty.
</math>|{{EquationRef|3}}}}

That is,

<math display="block">
    \Pr\!\left( \lim_{n\to\infty}\overline{X}_n = \mu \right) = 1.
</math>

What this means is that, as the number of trials ''n'' goes to infinity, the probability that the average of the observations converges to the expected value, is equal to one. The modern proof of the strong law is more complex than that of the weak law, and relies on passing to an appropriate sub-sequence.<ref name="TaoBlog" /> 

The strong law of large numbers can itself be seen as a special case of the [[Ergodic theory#Ergodic theorems|pointwise ergodic theorem]]. This view justifies the intuitive interpretation of the expected value (for Lebesgue integration only) of a random variable when sampled repeatedly as the "long-term average".  

Law 3 is called the strong law because random variables which converge strongly (almost surely) are guaranteed to converge weakly (in probability). However the weak law is known to hold in certain conditions where the strong law does not hold and then the convergence is only weak (in probability). See [[#Differences between the weak law and the strong law|Differences between the weak law and the strong law]].

The strong law applies to independent identically distributed random variables having an expected value (like the weak law). This was proved by Kolmogorov in 1930. It can also apply in other cases. Kolmogorov also showed, in 1933, that if the variables are independent and identically distributed, then for the average to converge almost surely on ''something'' (this can be considered another statement of the strong law), it is necessary that they have an expected value (and then of course the average will converge almost surely on that).<ref name=EMStrong>{{cite web|author1=Yuri Prokhorov| title=Strong law of large numbers|url=https://www.encyclopediaofmath.org/index.php/Strong_law_of_large_numbers| website=Encyclopedia of Mathematics}}</ref>

If the summands are independent but not identically distributed, then
{{NumBlk||<math display="block">
    \overline{X}_n - \operatorname{E}\big[\overline{X}_n\big]\ \overset{\text{a.s.}}{\longrightarrow}\ 0,
</math>|{{EquationRef|2}}}}

provided that each ''X''<sub>''k''</sub> has a finite second moment and

<math display="block">
    \sum_{k=1}^{\infty} \frac{1}{k^2} \operatorname{Var}[X_k] < \infty.
</math>

This statement is known as ''Kolmogorov's strong law'', see e.g. {{harvtxt|Sen|Singer|1993|loc=Theorem 2.3.10}}.

===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.)
}}

===Uniform laws of large numbers===
There are extensions of the law of large numbers to collections of estimators, where the convergence is uniform over the collection; thus the name ''uniform law of large numbers''.

Suppose ''f''(''x'',''θ'') is some [[Function (mathematics)|function]] defined for ''θ'' ∈ Θ, and continuous in ''θ''. Then for any fixed ''θ'', the sequence {''f''(''X''<sub>1</sub>,''θ''), ''f''(''X''<sub>2</sub>,''θ''), ...} will be a sequence of independent and identically distributed random variables, such that the sample mean of this sequence converges in probability to E[''f''(''X'',''θ'')]. This is the ''pointwise'' (in ''θ'') convergence.

A particular example of a '''uniform law of large numbers''' states the conditions under which the convergence happens ''uniformly'' in ''θ''. If<ref>{{harvnb|Newey|McFadden|1994|loc=Lemma 2.4}}</ref><ref>{{cite journal|doi=10.1214/aoms/1177697731|title=Asymptotic Properties of Non-Linear Least Squares Estimators|year=1969|last1=Jennrich|first1=Robert I.|journal=The Annals of Mathematical Statistics|volume=40|issue=2|pages=633–643|doi-access=free}}</ref>

# ''Θ'' is compact,
# ''f''(''x'',''θ'') is continuous at each ''θ'' ∈ Θ for [[Almost everywhere|almost all]] ''x''s, and measurable function of ''x'' at each ''θ''.
# there exists a [[Dominated convergence theorem|dominating]] function ''d''(''x'') such that E[''d''(''X'')] < ∞, and <math display="block"> \left\| f(x,\theta) \right\| \leq d(x) \quad\text{for all}\ \theta\in\Theta.</math>

Then E[''f''(''X'',''θ'')] is continuous in ''θ'', and

<math display="block">
    \sup_{\theta\in\Theta} \left\| \frac 1 n \sum_{i=1}^n f(X_i,\theta) - \operatorname{E}[f(X,\theta)] \right\| \overset{\mathrm{P}}{\rightarrow} \ 0.
  </math>

This result is useful to derive consistency of a large class of estimators (see [[Extremum estimator]]).

===Borel's law of large numbers===
'''Borel's law of large numbers''', named after [[Émile Borel]], states that if an experiment is repeated a large number of times, independently under identical conditions, then the proportion of times that any specified event is expected to occur approximately equals the probability of the event's occurrence on any particular trial; the larger the number of repetitions, the better the approximation tends to be. More precisely, if ''E'' denotes the event in question, ''p'' its probability of occurrence, and ''N<sub>n</sub>''(''E'') the number of times ''E'' occurs in the first ''n'' trials, then with probability one,<ref>{{cite journal | url=https://www.jstor.org/stable/2323947 | jstor=2323947 | doi=10.2307/2323947 | last1=Wen | first1=Liu | title=An Analytic Technique to Prove Borel's Strong Law of Large Numbers | journal=The American Mathematical Monthly | date=1991 | volume=98 | issue=2 | pages=146–148 }}</ref>
<math display="block"> \frac{N_n(E)}{n}\to p\text{ as }n\to\infty.</math>

This theorem makes rigorous the intuitive notion of probability as the expected long-run relative frequency of an event's occurrence.  It is a special case of any of several more general laws of large numbers in probability theory.

'''[[Chebyshev's inequality]]'''. Let ''X'' be a [[random variable]] with finite [[expected value]] ''μ'' and finite non-zero [[variance]] ''σ''<sup>2</sup>. Then for any [[real number]] {{math|''k'' > 0}},

<math display="block">
    \Pr(|X-\mu|\geq k\sigma) \leq \frac{1}{k^2}.
</math>

==Proof of the weak law==
Given ''X''<sub>1</sub>, ''X''<sub>2</sub>, ... an infinite sequence of [[i.i.d.]] random variables with finite expected value <math>E(X_1)=E(X_2)=\cdots=\mu<\infty</math>, we are interested in the convergence of the sample average

<math display="block">\overline{X}_n=\tfrac1n(X_1+\cdots+X_n). </math>

The weak law of large numbers states:
{{NumBlk||<math display="block">
    \overline{X}_n\ \overset{P}{\rightarrow}\ \mu \qquad\textrm{when}\ n \to \infty.
</math>|{{EquationRef|2}}}}

===Proof using Chebyshev's inequality assuming finite variance===
This proof uses the assumption of finite [[variance]] <math> \operatorname{Var} (X_i)=\sigma^2 </math> (for all <math>i</math>). The independence of the random variables implies no correlation between them, and we have that

<math display="block">
\operatorname{Var}(\overline{X}_n) = \operatorname{Var}(\tfrac1n(X_1+\cdots+X_n)) = \frac{1}{n^2} \operatorname{Var}(X_1+\cdots+X_n) = \frac{n\sigma^2}{n^2} = \frac{\sigma^2}{n}.
</math>

The common mean μ of the sequence is the mean of the sample average:

<math display="block">
E(\overline{X}_n) = \mu.
</math>

Using [[Chebyshev's inequality]] on <math>\overline{X}_n </math> results in

<math display="block">
\operatorname{P}( \left| \overline{X}_n-\mu \right| \geq \varepsilon) \leq \frac{\sigma^2}{n\varepsilon^2}.
</math>

This may be used to obtain the following:

<math display="block">
\operatorname{P}( \left| \overline{X}_n-\mu \right| < \varepsilon) = 1 - \operatorname{P}( \left| \overline{X}_n-\mu \right| \geq \varepsilon) \geq 1 - \frac{\sigma^2}{n \varepsilon^2 }.
</math>

As ''n'' approaches infinity, the expression approaches 1. And by definition of [[convergence in probability]], we have obtained
{{NumBlk||<math display="block">
    \overline{X}_n\ \overset{P}{\rightarrow}\ \mu \qquad\textrm{when}\ n \to \infty.
</math>|{{EquationRef|2}}}}

===Proof using convergence of characteristic functions===
By [[Taylor's theorem]] for [[complex function]]s, the [[Characteristic function (probability theory)|characteristic function]] of any random variable, ''X'', with finite mean μ, can be written as

<math display="block">\varphi_X(t) = 1 + it\mu + o(t), \quad t \rightarrow 0.</math>

All ''X''<sub>1</sub>, ''X''<sub>2</sub>, ... have the same characteristic function, so we will simply denote this ''φ''<sub>''X''</sub>.

Among the basic properties of characteristic functions there are

<math display="block">\varphi_{\frac 1 n X}(t)= \varphi_X(\tfrac t n) \quad \text{and} \quad
 \varphi_{X+Y}(t) = \varphi_X(t) \varphi_Y(t) \quad </math> if ''X'' and ''Y'' are independent.

These rules can be used to calculate the characteristic function of <math>\overline{X}_n</math> in terms of ''φ''<sub>''X''</sub>:

<math display="block">\varphi_{\overline{X}_n}(t)= \left[\varphi_X\left({t \over n}\right)\right]^n = \left[1 + i\mu{t \over n} + o\left({t \over n}\right)\right]^n \, \rightarrow \, e^{it\mu}, \quad \text{as} \quad n \to \infty.</math>

The limit ''e''<sup>''itμ''</sup> is the characteristic function of the constant random variable μ, and hence by the [[Lévy continuity theorem]], <math> \overline{X}_n</math> [[Convergence in distribution|converges in distribution]] to μ:

<math display="block">\overline{X}_n \, \overset{\mathcal D}{\rightarrow} \, \mu \qquad\text{for}\qquad n \to \infty.</math>

μ is a constant, which implies that convergence in distribution to μ and convergence in probability to μ are equivalent (see [[Convergence of random variables]].) Therefore,
{{NumBlk||<math display="block">
    \overline{X}_n\ \overset{P}{\rightarrow}\ \mu \qquad\textrm{when}\ n \to \infty.
</math>|{{EquationRef|2}}}}

This shows that the sample mean converges in probability to the derivative of the characteristic function at the origin, as long as the latter exists.

==Proof of the strong law==
We give a relatively simple proof of the strong law under the assumptions that the <math>X_i</math> are [[Independent and identically distributed random variables|iid]], <math> {\mathbb E}[X_i] =: \mu < \infty </math>, <math> \operatorname{Var} (X_i)=\sigma^2 < \infty</math>, and <math> {\mathbb E}[X_i^4] =: \tau < \infty </math>.

Let us first note that without loss of generality we can assume that <math>\mu = 0</math> by centering. In this case, the strong law says that

<math display="block">
    \Pr\!\left( \lim_{n\to\infty}\overline{X}_n = 0 \right) = 1,
</math>
or
<math display="block">
    \Pr\left(\omega: \lim_{n\to\infty}\frac{S_n(\omega)}n = 0 \right) = 1.
</math>
It is equivalent to show that
<math display="block">
    \Pr\left(\omega: \lim_{n\to\infty}\frac{S_n(\omega)}n \neq 0 \right) = 0,
</math>
Note that
<math display="block">
    \lim_{n\to\infty}\frac{S_n(\omega)}n \neq 0 \iff \exists\epsilon>0, \left|\frac{S_n(\omega)}n\right| \ge \epsilon\ \mbox{infinitely often},
</math>
and thus to prove the strong law we need to show that for every <math>\epsilon > 0</math>, we have
<math display="block">
    \Pr\left(\omega: |S_n(\omega)| \ge n\epsilon \mbox{ infinitely often} \right) = 0.
</math>
Define the events <math> A_n = \{\omega : |S_n| \ge n\epsilon\}</math>, and if we can show that 
<math display="block">
    \sum_{n=1}^\infty \Pr(A_n) <\infty,
</math>
then the Borel-Cantelli Lemma implies the result.  So let us estimate <math>\Pr(A_n)</math>.

We compute 
<math display="block">
    {\mathbb E}[S_n^4] = {\mathbb E}\left[\left(\sum_{i=1}^n X_i\right)^4\right] = {\mathbb E}\left[\sum_{1 \le i,j,k,l\le n} X_iX_jX_kX_l\right].
</math>
We first claim that every term of the form <math>X_i^3X_j, X_i^2X_jX_k, X_iX_jX_kX_l</math> where all subscripts are distinct, must have zero expectation.  This is because <math>{\mathbb E}[X_i^3X_j] = {\mathbb E}[X_i^3]{\mathbb E}[X_j]</math> by independence, and the last term is zero—and similarly for the other terms.  Therefore the only terms in the sum with nonzero expectation are <math>{\mathbb E}[X_i^4]</math> and <math>{\mathbb E}[X_i^2X_j^2]</math>.  Since the <math>X_i</math> are identically distributed, all of these are the same, and moreover <math>{\mathbb E}[X_i^2X_j^2]=({\mathbb E}[X_i^2])^2</math>.  

There are <math>n</math> terms of the form <math>{\mathbb E}[X_i^4]</math> and <math>3 n (n-1)</math> terms of the form <math>({\mathbb E}[X_i^2])^2</math>, and so
<math display="block">
    {\mathbb E}[S_n^4] = n \tau + 3n(n-1)\sigma^4.
</math>
Note that the right-hand side is a quadratic polynomial in <math>n</math>, and as such there exists a <math>C>0</math> such that <math> {\mathbb E}[S_n^4] \le Cn^2</math> for <math>n</math> sufficiently large.  By Markov,
<math display="block">
    \Pr(|S_n| \ge n \epsilon) \le \frac1{(n\epsilon)^4}{\mathbb E}[S_n^4] \le \frac{C}{\epsilon^4 n^2},
</math>
for <math>n</math> sufficiently large, and therefore this series is summable.  Since this holds for any <math>\epsilon > 0</math>, we have established the strong law of large numbers.<ref>Another proof was given by  {{cite journal |last1=Etemadi |first1=Nasrollah |title=An elementary proof of the strong law of large numbers |journal=Zeitschrift für Wahrscheinlichkeitstheorie und verwandte Gebiete |date=1981 |volume=55 |pages=119–122 |publisher=Springer|doi=10.1007/BF01013465 |s2cid=122166046 |doi-access=free }}</ref> The proof can be strengthened immensely by dropping all finiteness assumptions on the second and fourth moments. It can also be extended for example to discuss partial sums of distributions without any finite moments. Such proofs use more intricate arguments to prove the same Borel-Cantelli predicate, a strategy attributed to Kolmogorov to conceptually bring the limit inside the probability parentheses. <ref>For a proof without the added assumption of a finite fourth moment, see Section 22 of {{cite book|last = Billingsley | first = Patrick| title = Probability and Measure|date = 1979}}</ref>

== Consequences ==
The law of large numbers provides an expectation of an unknown distribution from a realization of the sequence, but also any feature of the [[probability distribution]].<ref name=":0" /> By applying [[Borel's law of large numbers]], one could easily obtain the probability mass function. For each event in the objective probability mass function, one could approximate the probability of the event's occurrence with the proportion of times that any specified event occurs. The larger the number of repetitions, the better the approximation. As for the continuous case: <math>C=(a-h,a+h]</math>, for small positive h. Thus, for large n:

<math display="block"> \frac{N_n(C)}{n}\thickapprox
p = P(X\in C) = \int_{a-h}^{a+h} f(x) \, dx 
\thickapprox
2hf(a)</math>

With this method, one can cover the whole x-axis with a grid (with grid size 2h) and obtain a bar graph which is called a [[histogram]].

== Applications ==
One application of the law of large numbers is an important method of approximation known as the [[Monte Carlo method]],<ref name=":1" /> which uses a random sampling of numbers to approximate numerical results. The algorithm to compute an integral of f(x) on an interval [a,b] is as follows:<ref name=":1" />

# Simulate uniform random variables X<sub>1</sub>, X<sub>2</sub>, ..., X<sub>n</sub> which can be done using a software, and use a random number table that gives U<sub>1</sub>, U<sub>2</sub>, ..., U<sub>n</sub> independent and identically distributed (i.i.d.) random variables on [0,1]. Then let X<sub>i</sub> = a+(b - a)U<sub>i</sub> for i= 1, 2, ..., n. Then X<sub>1</sub>, X<sub>2</sub>, ..., X<sub>n</sub> are independent and identically distributed uniform random variables on [a, b].
# Evaluate f(X<sub>1</sub>), f(X<sub>2</sub>), ..., f(X<sub>n</sub>)
# Take the average of f(X<sub>1</sub>), f(X<sub>2</sub>), ..., f(X<sub>n</sub>) by computing <math>(b-a)\tfrac{f(X_1)+f(X_2)+...+f(X_n)}{n}</math> and then by the strong law of large numbers, this converges to <math>(b-a)E(f(X_1))</math> = <math>(b-a)\int_{a}^{b} f(x)\tfrac{1}{b-a}{dx}</math> =<math>\int_{a}^{b} f(x){dx}</math>

We can find the integral of <math>f(x) = cos^2(x)\sqrt{x^3+1}</math> on [-1,2]. Using traditional methods to compute this integral is very difficult, so the Monte Carlo method can be used here.<ref name=":1" /> Using the above algorithm, we get

<math>\int_{-1}^{2} f(x){dx}</math>  = 0.905 when n=25

and

<math>\int_{-1}^{2} f(x){dx}</math>  = 1.028 when n=250

We observe that as n increases, the numerical value also increases. When we get the actual results for the integral we get

<math>\int_{-1}^{2} f(x){dx}</math> = 1.000194

When the LLN was used, the approximation of the integral was closer to its true value, and thus more accurate.<ref name=":1" />

Another example is the integration of <big>f(x) =</big> <math>\frac{e^x-1}{e-1}</math> on [0,1].<ref name=":2">{{Citation |last=Reiter |first=Detlev |title=The Monte Carlo Method, an Introduction |date=2008 |url=http://link.springer.com/10.1007/978-3-540-74686-7_3 |work=Computational Many-Particle Physics |series=Lecture Notes in Physics |volume=739 |pages=63–78 |editor-last=Fehske |editor-first=H. |access-date=2023-12-08 |place=Berlin, Heidelberg |publisher=Springer Berlin Heidelberg |language=en |doi=10.1007/978-3-540-74686-7_3 |isbn=978-3-540-74685-0 |editor2-last=Schneider |editor2-first=R. |editor3-last=Weiße |editor3-first=A.}}</ref> Using the Monte Carlo method and the LLN, we can see that as the number of samples increases, the numerical value gets closer to 0.4180233.<ref name=":2" />

==See also==
{{Div col|colwidth=20em}}
* [[Asymptotic equipartition property]]
* [[Central limit theorem]]
* [[Infinite monkey theorem]]
* [[A Treatise on Probability|Keynes' Treatise on Probability]]
* [[Law of averages]]
* [[Law of the iterated logarithm]]
* [[Law of truly large numbers]]
* [[Lindy effect]]
* [[Regression toward the mean]]
* [[Sortition]]
* [[Strong law of small numbers]]
{{Div col end}}

==Notes==
{{Reflist|2}}

==References==
{{refbegin}}
* {{cite book |last1=Grimmett |first1=G. R.  |last2=Stirzaker |first2=D. R. | title=Probability and Random Processes |edition=2nd | publisher=Clarendon Press |location=Oxford | year=1992 | isbn=0-19-853665-8}}
* {{cite book | first=Richard |last=Durrett | title=Probability: Theory and Examples |edition=2nd | publisher=Duxbury Press | year=1995}}
* {{cite book | author=Martin Jacobsen | publisher= HCØ-tryk |location=Copenhagen | year=1992|title=Videregående Sandsynlighedsregning |language=da |trans-title=Advanced Probability Theory |edition=3rd | isbn=87-91180-71-6}}
* {{cite book
  | last = Loève | first = Michel
  | title = Probability theory 1
  | year = 1977
  | edition = 4th
  | publisher = Springer
  }}
* {{cite book
  | last1 = Newey | first1 = Whitney K.
  | last2 = McFadden | first2 = Daniel | author-link2 = Daniel McFadden
  | title = Large sample estimation and hypothesis testing
  | series = Handbook of econometrics |volume=IV |chapter=36
  | year = 1994
  | publisher = Elsevier Science
  | pages = 2111–2245
  }}
* {{cite book
  | last = Ross | first = Sheldon
  | title = A first course in probability
  | year = 2009
  | edition = 8th
  | publisher = Prentice Hall
  | isbn = 978-0-13-603313-4
  }}
* {{cite book
  | last1 = Sen | first1 = P. K
  | last2 = Singer | first2 = J. M.
  | year = 1993
  | title = Large sample methods in statistics
  | publisher = Chapman & Hall
  }}
* {{cite journal|author1-link=Eugene Seneta|last=Seneta|first=Eugene|title=A Tricentenary history of the Law of Large Numbers| journal=Bernoulli| volume=19| issue=4| pages=1088–1121| date=2013|doi=10.3150/12-BEJSP12|arxiv=1309.6488|s2cid=88520834}}

{{refend}}

==External links==
* {{springer|title=Law of large numbers|id=p/l057720}}
* {{MathWorld|urlname=WeakLawofLargeNumbers|title=Weak Law of Large Numbers}}
* {{MathWorld|urlname=StrongLawofLargeNumbers|title=Strong Law of Large Numbers}}
* [https://web.archive.org/web/20081110071309/http://animation.yihui.name/prob:law_of_large_numbers Animations for the Law of Large Numbers] by Yihui Xie using the [[R (programming language)|R]] package [https://cran.r-project.org/package=animation animation]
* [http://www.businessinsider.com/law-of-large-numbers-tim-cook-2015-2 Apple CEO Tim Cook said something that would make statisticians cringe]. "We don't believe in such laws as laws of large numbers. This is sort of, uh, old dogma, I think, that was cooked up by somebody [..]" said Tim Cook and while: "However, the law of large numbers has nothing to do with large companies, large revenues, or large growth rates. The law of large numbers is a fundamental concept in probability theory and statistics, tying together theoretical probabilities that we can calculate to the actual outcomes of experiments that we empirically perform.'' explained [[Business Insider]]''
{{Authority control}}

[[Category:Theorems in probability theory]]
[[Category:Mathematical proofs]]
[[Category:Asymptotic theory (statistics)]]
[[Category:Theorems in statistics]]
[[Category:Large numbers]]