Difference between revisions of "Borel strong law of large numbers"
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Historically, the first variant of the [[Strong law of large numbers|strong law of large numbers]], formulated and proved by E. Borel [[#References|[1]]] 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 [[#References|[1]]] showed that | Historically, the first variant of the [[Strong law of large numbers|strong law of large numbers]], formulated and proved by E. Borel [[#References|[1]]] 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 [[#References|[1]]] showed that | ||
Revision as of 19:09, 29 January 2012
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 [1] in the context of the Bernoulli scheme (cf. Bernoulli trials). Consider independent random variables 
which are identically distributed and assume one of two values 0 and 1 with probability of 1/2 each; the expression 
will then give the number of successful trials in a Bernoulli scheme in which the probability of success is 1/2. Borel [1] showed that
 |
with probability one as 
. It was subsequently (1914) shown by G.H. Hardy and J.E. Littlewood that, almost certainly,
 |
after which (1922) the stronger result:
 |
was proved by A.Ya. Khinchin. See also Law of the iterated logarithm.
References
| [1] | E. Borel, "Les probabilités dénombrables et leurs applications arithmetique" Rend. Circ. Mat. Palermo (2) , 27 (1909) pp. 247–271 |
| [2] | M. Kac, "Statistical independence in probability, analysis and number theory" , Math. Assoc. Amer. (1963) |
Borel strong law of large numbers. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Borel_strong_law_of_large_numbers&oldid=13249