Borel strong law of large numbers

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 [B] in the context of the Bernoulli scheme (cf. Bernoulli trials). Consider independent random variables $X_1,\ldots,X_n,\ldots$ which are identically distributed and assume one of two values 0 and 1 with probability of 1/2 each; the expression $S_n = \sum_{k=1}^n X_k$ will then give the number of successful trials in a Bernoulli scheme in which the probability of success is 1/2. Borel [B] showed that $$ \frac{S_n}{n} \rightarrow \frac12 $$ with probability one as $n \rightarrow \infty$. It was subsequently (1914) shown by G.H. Hardy and J.E. Littlewood that, almost certainly, $$ \limsup_{n \rightarrow \infty} \frac{ \left\vert{ \frac{S_n}{n} - \frac12 }\right\vert }{ \sqrt{n \log n} } < \frac{1}{\sqrt2} $$ after which (1922) the stronger result: $$ \mathrm{Prob}\left[{ \limsup_{n \rightarrow \infty} \frac{ \left\vert{ \frac{S_n}{n} - \frac12 }\right\vert }{ \sqrt{n \log\log n} } = \frac{1}{\sqrt2} }\right] = 1 $$ was proved by A.Ya. Khinchin. See also Law of the iterated logarithm.

References

[B]	E. Borel, "Les probabilités dénombrables et leurs applications arithmetique" Rend. Circ. Mat. Palermo (2) , 27 (1909) pp. 247–271
[K]	M. Kac, "Statistical independence in probability, analysis and number theory" , Math. Assoc. Amer. (1963) MR1530983 MR0110114 Zbl 0112.09101