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{{MSC|60F10|60F15}}

[[Category:Limit theorems]]

A limit theorem in probability theory which is a refinement of the [[Strong law of large numbers|strong law of large numbers]]. Let  $  X _ {1} , X _ {2} \dots $
be a sequence of random variables and let

$$ 
S _ {n}  =  X _ {1} + \dots + X _ {n} .
$$

For simplicity one assumes that  $  S _ {n} $
has zero median for each  $  n $. 
While the theorem on the strong law of large numbers deals with conditions under which  $  S _ {n} /a _ {n} \rightarrow 0 $
almost surely ( $  a.s. $) 
for  $  n \rightarrow \infty $, 
where  $  \{ a _ {n} \} $
is a sequence of numbers, the theorem on the law of the iterated logarithm deals with sequences of numbers  $  \{ c _ {n} \} $
such that

$$ \tag{1 }
\lim\limits _ {n \rightarrow \infty }  \sup  
\frac{S _ {n} }{c _ {n} }

 =  1 \  ( a.s.)
$$

or

$$ \tag{2 }
\lim\limits _ {n \rightarrow \infty }  \sup  
\frac{| S _ {n} | }{c _ {n} }

 =  1 \  ( a.s.).
$$

Relation (1) is equivalent to

$$ 
{\mathsf P} \{ S _ {n} > ( 1 + \epsilon ) c _ {n}  ( i.o.) \}  =  0
$$

and

$$ 
{\mathsf P} \{ S _ {n} > ( 1 - \epsilon ) c _ {n}  ( i.o.) \}  =  1
$$

for any  $  \epsilon > 0 $, 
where  $  i.o. $
denotes infinitely often.

Relations of the form of (1) and (2) hold under more restrictive conditions than the estimates implied by the strong law of large numbers. If  $  \{ X _ {n} \} $
is a sequence of independent random variables having identical distributions with mathematical expectations equal to zero, then

$$ 

\frac{S _ {n} }{n}
 \rightarrow 0 \  \textrm{ (a.s.)  } \  \textrm{ for }  n \rightarrow \infty
$$

(Kolmogorov's theorem); if the additional condition  $  0 < {\mathsf E} X _ {1}  ^ {2} < \infty $
is satisfied, then one has the stronger relation (2), in which

$$ 
c _ {n}  =  ( 2nb  { \mathop{\rm ln}   \mathop{\rm ln} } ( nb))  ^ {1/2} ,
$$

where  $  b = {\mathsf E} X _ {1}  ^ {2} $(
the Hartman–Wintner theorem).

The first theorem of general type on the law of the iterated logarithm was the following result obtained by A.N. Kolmogorov {{Cite|Ko}}. Let  $  \{ X _ {n} \} $
be a sequence of independent random variables with mathematical expectations equal to zero and with finite variances, and let

$$ 
B _ {n}  =  \sum_{k=1}^ { n }  {\mathsf E} X _ {k}  ^ {2} .
$$

If  $  B _ {n} \rightarrow \infty $
for  $  n \rightarrow \infty $
and if there exists a sequence of positive constants  $  \{ M _ {n} \} $
such that

$$ 
| X _ {n} |  \leq   M _ {n} \  
\textrm{ and } \ \ 
M _ {n}  =  o \left ( \left ( 
\frac{B _ {n} }{ \mathop{\rm ln}   \mathop{\rm ln}  B _ {n} }
 \right )
 ^ {1/2} \right ) ,
$$

then (1) and (2) are satisfied for

$$ 
c _ {n}  =  ( 2B _ {n}   \mathop{\rm ln}   \mathop{\rm ln}  B _ {n} )  ^ {1/2} .
$$

In the particular case where  $  \{ X _ {n} \} $
is a sequence of independent random variables having identical distributions with two possible values, this assertion was derived by A.Ya. Khinchin {{Cite|Kh}}. J. Marcinkiewicz and A. Zygmund {{Cite|MZ}} showed that under the conditions of Kolmogorov's theorem one cannot replace  $  o $
by  $  O $. 
W. Feller {{Cite|F}} examined a generalization of Kolmogorov's law of the iterated logarithm for sequences of independent bounded non-identically distributed random variables. See {{Cite|S}} for other generalizations of the law; there is also the following result (see {{Cite|S2}}), which is related to the Hartman–Wintner theorem: If  $  \{ X _ {n} \} $
is a sequence of independent random variables having identical distributions with infinite variances, then

$$ 
\lim\limits _ {n \rightarrow \infty }  \sup  
\frac{| S _ {n} | }{( n   \mathop{\rm ln}  n   \mathop{\rm ln}  n)  ^ {1/2} }
  =  \infty \  ( a.s.).
$$

The results obtained on the law of the iterated logarithm for sequences of independent random variables have served as a starting point for numerous researches on the applicability of this law to sequences of dependent random variables and vectors and to random processes.

====References====
{|
|valign="top"|{{Ref|Ko}}|| A.N. Kolmogoroff, "Ueber das Gesetz des iterierten Logarithmus" ''Math. Ann.'' , '''101''' (1929) pp. 126–135
|-
|valign="top"|{{Ref|Kh}}|| A. Khintchine, "Ueber einen Satz der Wahrscheinlichkeitsrechnung" ''Fund. Math.'' , '''6''' (1924) pp. 9–20 {{MR|}} {{ZBL|50.0344.02}}
|-
|valign="top"|{{Ref|MZ}}|| J. Marcinkiewicz, A. Zygmund, "Rémarque sur la loi du logarithme itéré" ''Fund. Math.'' , '''29''' (1937) pp. 215–222 {{MR|}} {{ZBL|0018.03204}} {{ZBL|63.1076.03}}
|-
|valign="top"|{{Ref|F}}|| W. Feller, "The general form of the so-called law of the iterated logarithm" ''Trans. Amer. Math. Soc.'' , '''54''' (1943) pp. 373–402 {{MR|0009263}} {{ZBL|0063.08417}}
|-
|valign="top"|{{Ref|S}}|| V. Strassen, "An invariance principle for the law of the iterated logarithm" ''Z. Wahrsch. Verw. Geb.'' , '''3''' (1964) pp. 211–226 {{MR|0175194}} {{ZBL|0132.12903}}
|-
|valign="top"|{{Ref|S2}}|| V. Strassen, "A converse to the law of iterated logarithm" ''Z. Wahrsch. Verw. Geb.'' , '''4''' (1965–1966) pp. 265–268 {{MR|}} {{ZBL|0141.16501}}
|-
|valign="top"|{{Ref|HW}}|| P. Hartman, A. Wintner, "On the law of the iterated logarithm" ''Amer. J. Math.'' , '''63''' (1941) pp. 169–176 {{MR|0003497}} {{ZBL|0024.15802}} {{ZBL|67.0460.03}}
|-
|valign="top"|{{Ref|La}}|| J. Lamperty, "Probability" , Benjamin (1966)
|-
|valign="top"|{{Ref|P}}|| V.V. Petrov, "Sums of independent random variables" , Springer (1975) (Translated from Russian) {{MR|0388499}} {{ZBL|0322.60043}} {{ZBL|0322.60042}}
|-
|valign="top"|{{Ref|HH}}|| P. Hall, C.C. Heyde, "Martingale limit theory and its application" , Acad. Press (1980) {{MR|0624435}} {{ZBL|0462.60045}}
|-
|valign="top"|{{Ref|F2}}|| W. Feller, "An introduction to probability theory and its applications" , '''1''' , Wiley (1968) {{MR|0228020}} {{ZBL|0155.23101}}
|-
|valign="top"|{{Ref|Lo}}|| M. Loève, "Probability theory" , Princeton Univ. Press (1963) pp. Chapt. XIV {{MR|0203748}} {{ZBL|0108.14202}}
|}