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Difference between revisions of "Borel strong law of large numbers"

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[[Category:Limit theorems]]
 
[[Category:Limit theorems]]
  
Historically, the first variant of the [[Strong law of large numbers|strong law of large numbers]], formulated and proved by E. Borel {{Cite|B}} in the context of the Bernoulli scheme (cf. [[Bernoulli trials|Bernoulli trials]]). Consider independent random variables <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017150/b0171501.png" /> which are identically distributed and assume one of two values 0 and 1 with probability of 1/2 each; the expression <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017150/b0171502.png" /> will then give the number of successful trials in a Bernoulli scheme in which the probability of success is 1/2. Borel {{Cite|B}} showed that
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Historically, the first variant of the [[strong law of large numbers]], formulated and proved by E. Borel {{Cite|B}} in the context of the Bernoulli scheme (cf. [[Bernoulli trials]]). Consider independent random variables $X_1,\ldots,X_n,\ldots$ which are identically distributed and assume one of two values 0 and 1 with probability of 1/2 each; the expression $S_n = \sum_{k=1}^n X_k$ will then give the number of successful trials in a Bernoulli scheme in which the probability of success is 1/2. Borel {{Cite|B}} showed that
 
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$$
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017150/b0171503.png" /></td> </tr></table>
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\frac{S_n}{n} \rightarrow \frac12
 
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$$
with probability one as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017150/b0171504.png" />. It was subsequently (1914) shown by G.H. Hardy and J.E. Littlewood that, almost certainly,
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with probability one as $n \rightarrow \infty$. It was subsequently (1914) shown by G.H. Hardy and J.E. Littlewood that, almost certainly,
 
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$$
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017150/b0171505.png" /></td> </tr></table>
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\limsup_{n \rightarrow \infty} \frac{ \left\vert{ \frac{S_n}{n} - \frac12 }\right\vert }{ \sqrt{n \log n} } < \frac{1}{\sqrt2}
 
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$$
 
after which (1922) the stronger result:
 
after which (1922) the stronger result:
 
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$$
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017150/b0171506.png" /></td> </tr></table>
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\mathrm{Prob}\left[{ \limsup_{n \rightarrow \infty} \frac{ \left\vert{ \frac{S_n}{n} - \frac12 }\right\vert }{ \sqrt{n \log\log n} } = \frac{1}{\sqrt2} }\right] = 1
 
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$$
 
was proved by A.Ya. Khinchin. See also [[Law of the iterated logarithm|Law of the iterated logarithm]].
 
was proved by A.Ya. Khinchin. See also [[Law of the iterated logarithm|Law of the iterated logarithm]].
  
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|valign="top"|{{Ref|K}}|| M. Kac, "Statistical independence in probability, analysis and number theory" , Math. Assoc. Amer. (1963) {{MR|1530983}} {{MR|0110114}} {{ZBL|0112.09101}}
 
|valign="top"|{{Ref|K}}|| M. Kac, "Statistical independence in probability, analysis and number theory" , Math. Assoc. Amer. (1963) {{MR|1530983}} {{MR|0110114}} {{ZBL|0112.09101}}
 
|}
 
|}
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Latest revision as of 10:23, 8 February 2017

2020 Mathematics Subject Classification: Primary: 60F15 [MSN][ZBL]

Historically, the first variant of the strong law of large numbers, formulated and proved by E. Borel [B] in the context of the Bernoulli scheme (cf. Bernoulli trials). Consider independent random variables $X_1,\ldots,X_n,\ldots$ which are identically distributed and assume one of two values 0 and 1 with probability of 1/2 each; the expression $S_n = \sum_{k=1}^n X_k$ will then give the number of successful trials in a Bernoulli scheme in which the probability of success is 1/2. Borel [B] showed that $$ \frac{S_n}{n} \rightarrow \frac12 $$ with probability one as $n \rightarrow \infty$. It was subsequently (1914) shown by G.H. Hardy and J.E. Littlewood that, almost certainly, $$ \limsup_{n \rightarrow \infty} \frac{ \left\vert{ \frac{S_n}{n} - \frac12 }\right\vert }{ \sqrt{n \log n} } < \frac{1}{\sqrt2} $$ after which (1922) the stronger result: $$ \mathrm{Prob}\left[{ \limsup_{n \rightarrow \infty} \frac{ \left\vert{ \frac{S_n}{n} - \frac12 }\right\vert }{ \sqrt{n \log\log n} } = \frac{1}{\sqrt2} }\right] = 1 $$ was proved by A.Ya. Khinchin. See also Law of the iterated logarithm.

References

[B] E. Borel, "Les probabilités dénombrables et leurs applications arithmetique" Rend. Circ. Mat. Palermo (2) , 27 (1909) pp. 247–271
[K] M. Kac, "Statistical independence in probability, analysis and number theory" , Math. Assoc. Amer. (1963) MR1530983 MR0110114 Zbl 0112.09101
How to Cite This Entry:
Borel strong law of large numbers. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Borel_strong_law_of_large_numbers&oldid=40204
This article was adapted from an original article by A.V. Prokhorov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article