Llei dels grans nombres: diferència entre les revisions

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.

 

 

==Formes==

La hipòtesi de [[variància]] finita Var(''X''₁) = Var(''X''₂) = ... = ''σ''² < ∞ '''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 diferència entre la versió de la llei feble i la forta rau en la manera en què la convergència és definida.

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.

 

 

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).

 

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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>

 

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The strong law of large numbers can itself be seen as a special case of the [[Ergodic theory#Ergodic theorems|pointwise ergodic theorem]].

 

 

If the summands are independent but not identically distributed, then

An example of a series where the weak law applies but not the strong law is when ''X_k'' 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_k'' 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.

 

 

===Differences between the weak law and the strong law===

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>

 

 

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> F(x)=\frac{e}{-2x\ln(-x)},x \le -e </math>

 

===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''₁,''θ''), ''f''(''X''₂,''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.

 

 

# Θ is compact,

==References==

{{refbegin}}

| ref = CITEREFLo.C3.A8ve1977

}}

| series = Handbook of econometrics, vol. IV, Ch. 36

| ref = CITEREFNeweyMcFadden1994

}}

| isbn = 978-0-13-603313-4

}}

| ref = CITEREFSenSinger1993

}}

 

{{refend}}