Namespaces
Variants
Actions

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

From Encyclopedia of Mathematics
Jump to: navigation, search
(Importing text file)
 
(60F15)
Line 1: Line 1:
 +
{{MSC|60F15}}
 +
 +
[[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 [[#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)
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=20797
This article was adapted from an original article by A.V. Prokhorov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article