Editing Law of large numbers (section)

==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}}