Difference between revisions of "Borel-Cantelli lemma"
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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 <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 | ||
Revision as of 18:50, 30 January 2012
2020 Mathematics Subject Classification: Primary: 60-01 Secondary: 60F1560F20 [MSN][ZBL]
A frequently used statement on infinite sequences of random events. Let 
be a sequence of events from a certain probability space and let 
be the event consisting in the occurance of (only) a finite number out of the events 
, 
. 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
| [1] | E. Borel, "Les probabilités dénombrables et leurs applications arithmetique" Rend. Circ. Mat. Palermo (2) , 27 (1909) pp. 247–271 |
| [2] | F.P. Cantelli, "Sulla probabilità come limite della frequenza" Atti Accad. Naz. Lincei , 26 : 1 (1917) pp. 39–45 |
| [3] | M. Loève, "Probability theory" , Princeton Univ. Press (1963) |
Comments
The Borel–Cantelli lemma can be used in number theory to prove the so-called "normality" of almost-all natural numbers, cf. [a1], Chapt. 8, Sect. 6.
References
| [a1] | W. Feller, "An introduction to probability theory and its applications" , 1 , Wiley (1957) pp. Chapt.14 |
Borel-Cantelli lemma. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Borel-Cantelli_lemma&oldid=19142