View source for Borel-Cantelli lemma

{{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

<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>

then

<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/b0170406.png" /></td> </tr></table>

If the events <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017040/b0170407.png" /> are mutually independent, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017040/b0170408.png" /> or 0, depending on whether the series <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017040/b0170409.png" /> converges or diverges, i.e. in this case the condition (*) is necessary and sufficient for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017040/b01704010.png" />; this is the so-called Borel criterion for  "zero or one"  (cf. [[Zero-one law|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|strong law of large numbers]].

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  E. Borel,   "Les probabilités dénombrables et leurs applications arithmetique"  ''Rend. Circ. Mat. Palermo (2)'' , '''27'''  (1909)  pp. 247–271</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  F.P. Cantelli,   "Sulla probabilità come limite della frequenza"  ''Atti Accad. Naz. Lincei'' , '''26''' :  1  (1917)  pp. 39–45</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  M. Loève,   "Probability theory" , Princeton Univ. Press  (1963)</TD></TR></table>



====Comments====
The Borel–Cantelli lemma can be used in number theory to prove the so-called  "normality"  of almost-all natural numbers, cf. [[#References|[a1]]], Chapt. 8, Sect. 6.

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  W. Feller,   "An introduction to probability theory and its applications" , '''1''' , Wiley  (1957)  pp. Chapt.14</TD></TR></table>