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 which are identically distributed and assume one of two values 0 and 1 with probability of 1/2 each; the expression 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

with probability one as . It was subsequently (1914) shown by G.H. Hardy and J.E. Littlewood that, almost certainly,

after which (1922) the stronger result:

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