Llei dels grans nombres: diferència entre les revisions
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El matemàtic italià [[Gerolamo Cardano]] (1501–1576) va constatar, sense demostrar, que la precisió de les mesures estadístiques empíriques tendien a millorar amb el nombre d'assajos<ref>Mlodinow, L. ''The Drunkard's Walk.'' New York: Random House, 2008. p. 50.</ref> i es va formalitzar com a llei dels grans nombres. Una forma especial d'aquesta llei, per les variables aleatòries binàries) va ser demostrar primerament per [[Jacob Bernoulli]].<ref>Jakob Bernoulli, ''Ars Conjectandi: Usum & Applicationem Praecedentis Doctrinae in Civilibus, Moralibus & Oeconomicis'', 1713, Chapter 4, (Translated into English by Oscar Sheynin)</ref> Li va costar més de 20 anys desenvolupar una prova matemàtica suficientment rigorosa, que va ser publicada en la seva obra ''Ars Conjectandi'' (L'art de la conjentura) el 1713. El va anomenar el seu "Teorema daurat", però va ser conegut generalment amb el nom del "Teorema de Bernoulli", que no s'ha de confondre amb el [[Principi de Bernoulli]], referit al seu cosí [[Daniel Bernoulli]]. El 1837, [[Siméon Denis Poisson|S.D. Poisson]] el va desenvolupar més sota el nom "''la loi des grands nombres''" ("La llei dels grans nombres").<ref>Poisson names the "law of large numbers" (''la loi des grands nombres'') in: S.D. Poisson, ''Probabilité des jugements en matière criminelle et en matière civile, précédées des règles générales du calcul des probabilitiés'' (Paris, France: Bachelier, 1837), [https://books.google.com/books?id=uovoFE3gt2EC&pg=PA7#v=onepage p. 7]. He attempts a two-part proof of the law on pp. 139–143 and pp. 277 ff.</ref><ref>Hacking, Ian. (1983) "19th-century Cracks in the Concept of Determinism", ''Journal of the History of Ideas'', 44 (3), 455-475 {{jstor|2709176}}</ref> D'aleshores encà, es coneix pels dos noms, tot i que la "Llei dels grans nombres" és més freqüent.
Després que Benroulli i Poisson publiquessin els seus esforços, altres matemàtics van contribuir a refinar la llei, com [[Pafnuty Chebyshev|Chebyshev]],<ref>{{
==Formes==
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La hipòtesi de [[variància]] finita Var(''X''<sub>1</sub>) = Var(''X''<sub>2</sub>) = ... = ''σ''<sup>2</sup> < ∞ '''no és necessària'''. Una variància gran o infinita farà que la convergència sigui més lenta, però la llei dels grans nombres es manté de tota manera. Aquesta hipòtesi, però, sovint s'usa per tal de fer la demostració més fàcil i més curta.
La [[Independència estadística|independència]] mútua de les variables aleatòries pot ser substituïda per la independència parell a en ambdues versoins de la llei.<ref>{{
La diferència entre la versió de la llei feble i la forta rau en la manera en què la convergència és definida.
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What this means is that the probability that, as the number of trials ''n'' goes to infinity, the average of the observations converges to the expected value, is equal to one.
The proof is more complex than that of the weak law.<ref>{{
Almost sure convergence is also called strong convergence of random variables. This version is called the strong law because random variables which converge strongly (almost surely) are guaranteed to converge weakly (in probability). The strong law implies the weak law but not vice versa, when the strong law conditions hold the variable converges both strongly (almost surely) and 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).{{clarify|reason=Warning: This is ambiguous. By using the fuzzy word "may", it says that EITHER the weak law is known to hold under broader conditions than the strong law, OR the weak law is known to hold under certain conditions where the strong law is not known to hold, OR it is not known whether the weak law holds under broader conditions that the strong law. |date=
<!--The reference provided for the following doesn't seem to say this:
There are different views among mathematicians whether the two laws could be unified to one law, thereby replacing the weak law.<ref name="Testing Statistical Hypotheses">{{cite book|url=https://books.google.com/books?id=-kzPBAAAQBAJ&pg=PA219&lpg=PA219#v=onepage&q=is%20of%20very%20limited%20interest%20and%20should%20be%20replaced%20by%20the%20more%20precise%20and%20more%20useful%20strong%20law%20of%20large%20numbers | title=Law of large numbers views }}</ref>
To date it has not been possible to prove that the strong law conditions are the same as those of the weak law.{{citation needed|date=
-->
The strong law of large numbers can itself be seen as a special case of the [[Ergodic theory#Ergodic theorems|pointwise ergodic theorem]].
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>{{
If the summands are independent but not identically distributed, then
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An example of a series where the weak law applies but not the strong law is when ''X<sub>k</sub>'' is plus or minus <math>\sqrt{k/\log\log\log k}</math> (starting at sufficiently large ''k'' so that the denominator is positive) with probability 1/2 for each.<ref name=EMStrong/> The variance of ''X<sub>k</sub>'' is then <math>k/\log\log\log k.</math> Kolmogorov's strong law does not apply because the partial sum in his criterion up to ''k=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>\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>1/\sqrt{2\log\log\log n}</math>), but for a given ε, there is probability which does not go to zero with ''n'' that the average sometime after the ''n''th trial will come back up to ε. Since this probability does not go to zero {{clarify|reason=How can we see this does not go to zero? |date=
===Differences between the weak law and the strong law===
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The ''strong law'' shows that this [[almost surely]] will not occur. In particular, it implies that with probability 1, we have that for any {{nowrap|''ε'' > 0}} the inequality <math style="vertical-align:-.4em">|\overline{X}_n -\mu| < \varepsilon</math> holds for all large enough ''n''.<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">{{
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:
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:<math> F(x)=\frac{e}{-2x\ln(-x)},x \le -e </math>
:then it has no expected value, but the weak law is true.<ref>{{
===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.
The '''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>{{
# Θ is compact,
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==References==
{{refbegin}}
* {{
* {{
* {{
▲ | last = Loève | first = Michel
|any= 1977
▲ | title = Probability theory 1
|
▲ | publisher = Springer Verlag
| ref = CITEREFLo.C3.A8ve1977
}}
▲ | last1 = Newey | first1 = Whitney K.
▲ | last2 = McFadden | first2 = Daniel | authorlink2 = Daniel McFadden
▲ | title = Large sample estimation and hypothesis testing
| series = Handbook of econometrics, vol. IV, Ch. 36
|
|
|
| ref = CITEREFNeweyMcFadden1994
}}
* {{ref-llibre|cognom= Ross |nom= Sheldon
|
|any= 2009
|
▲ | publisher = Prentice Hall press
| isbn = 978-0-13-603313-4
}}
* {{ref-llibre|cognom1= Sen |nom1= P. K
|
▲ | year = 1993
▲ | title = Large sample methods in statistics
▲ | publisher = Chapman & Hall, Inc
| ref = CITEREFSenSinger1993
}}
* {{
{{refend}}
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