The law of the iterated logarithm gives the almost-sure asymptotic magnitude of the fluctuations of a sum of independent random variables.
If 
,
,
... are independent and identically
distributed with expectation value 0 and
finite positive variance 
, and 
, then
 |
almost surely. Applying the result to 
gives the corresponding infimum
limit 
.
The theorem refines the law of large numbers: the latter gives the scale on which 
tends to 0, while the iterated logarithm gives the precise
envelope of the remaining fluctuations.
