View source for Borel strong law of large numbers

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

<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/b017150/b0171503.png" /></td> </tr></table>

with probability one as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017150/b0171504.png" />. It was subsequently (1914) shown by G.H. Hardy and J.E. Littlewood that, almost certainly,

<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/b017150/b0171505.png" /></td> </tr></table>

after which (1922) the stronger result:

<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/b017150/b0171506.png" /></td> </tr></table>

was proved by A.Ya. Khinchin. See also [[Law of the iterated logarithm|Law of the iterated logarithm]].

====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">  M. Kac,   "Statistical independence in probability, analysis and number theory" , Math. Assoc. Amer.  (1963)</TD></TR></table>