Namespaces
Variants
Actions

Difference between revisions of "Borel-Cantelli lemma"

From Encyclopedia of Mathematics
Jump to: navigation, search
(Undo revision 33503 by Fochsner (talk) see talk)
m (TeX)
Line 1: Line 1:
 
{{MSC|60-01|60F15,60F20}}
 
{{MSC|60-01|60F15,60F20}}
  
A frequently used statement on infinite sequences of random events. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017040/b0170401.png" /> be a sequence of events from a certain probability space and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017040/b0170402.png" /> be the event consisting in the occurance of (only) a finite number out of the events <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017040/b0170403.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017040/b0170404.png" />. Then, according to the Borel–Cantelli lemma, if
+
A frequently used statement on infinite sequences of random events. Let $A_1,\dots, A_n, \dots$ be a sequence of events from a certain probability space and let $A$ be the event consisting in the occurance of (only) a finite number out of the events $A_n$, $n=1,2\dots$. Then, according to the Borel–Cantelli lemma, if
  
 
<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/b017040/b0170405.png" /></td> <td valign="top" style="width:5%;text-align:right;">(*)</td></tr></table>
 
<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/b017040/b0170405.png" /></td> <td valign="top" style="width:5%;text-align:right;">(*)</td></tr></table>

Revision as of 11:27, 8 October 2014

2010 Mathematics Subject Classification: Primary: 60-01 Secondary: 60F1560F20 [MSN][ZBL]

A frequently used statement on infinite sequences of random events. Let $A_1,\dots, A_n, \dots$ be a sequence of events from a certain probability space and let $A$ be the event consisting in the occurance of (only) a finite number out of the events $A_n$, $n=1,2\dots$. Then, according to the Borel–Cantelli lemma, if

(*)

then

If the events are mutually independent, then or 0, depending on whether the series converges or diverges, i.e. in this case the condition (*) is necessary and sufficient for ; this is the so-called Borel criterion for "zero or one" (cf. Zero-one law). This last criterion can be generalized to include certain classes of dependent events. The Borel–Cantelli lemma is used, for example, to prove the strong law of large numbers.

References

[B] E. Borel, "Les probabilités dénombrables et leurs applications arithmetiques" Rend. Circ. Mat. Palermo (2) , 27 (1909) pp. 247–271 Zbl 40.0283.01
[C] F.P. Cantelli, "Sulla probabilità come limite della frequenza" Atti Accad. Naz. Lincei , 26 : 1 (1917) pp. 39–45 Zbl 46.0779.02
[L] M. Loève, "Probability theory" , Princeton Univ. Press (1963) MR0203748 Zbl 0108.14202

Comments

The Borel–Cantelli lemma can be used in number theory to prove the so-called "normality" of almost-all natural numbers, cf. [F], Chapt. 8, Sect. 6.

References

[F] W. Feller, "An introduction to probability theory and its applications", 1 , Wiley (1957) pp. Chapt.14
How to Cite This Entry:
Borel-Cantelli lemma. Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Borel-Cantelli_lemma&oldid=33512
This article was adapted from an original article by A.V. Prokhorov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article