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A limit theorem in probability theory which is a refinement of the [[Strong law of large numbers|strong law of large numbers]]. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057740/l0577401.png" /> be a sequence of random variables and let
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<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/l/l057/l057740/l0577402.png" /></td> </tr></table>
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{{TEX|auto}}
 +
{{TEX|done}}
  
For simplicity one assumes that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057740/l0577403.png" /> has zero median for each <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057740/l0577404.png" />. While the theorem on the strong law of large numbers deals with conditions under which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057740/l0577405.png" /> almost surely (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057740/l0577406.png" />) for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057740/l0577407.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057740/l0577408.png" /> is a sequence of numbers, the theorem on the law of the iterated logarithm deals with sequences of numbers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057740/l0577409.png" /> such that
+
{{MSC|60F10|60F15}}
  
<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/l/l057/l057740/l05774010.png" /></td> <td valign="top" style="width:5%;text-align:right;">(1)</td></tr></table>
+
[[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
 
or
  
<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/l/l057/l057740/l05774011.png" /></td> <td valign="top" style="width:5%;text-align:right;">(2)</td></tr></table>
+
$$ \tag{2 }
 +
\lim\limits _ {n \rightarrow \infty }  \sup 
 +
\frac{| S _ {n} | }{c _ {n} }
 +
 
 +
= 1 \  ( a.s.).
 +
$$
  
 
Relation (1) is equivalent to
 
Relation (1) is equivalent to
  
<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/l/l057/l057740/l05774012.png" /></td> </tr></table>
+
$$
 +
{\mathsf P} \{ S _ {n} > ( 1 + \epsilon ) c _ {n}  ( i.o.) \}  = 0
 +
$$
  
 
and
 
and
  
<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/l/l057/l057740/l05774013.png" /></td> </tr></table>
+
$$
 +
{\mathsf P} \{ S _ {n} > ( 1 - \epsilon ) c _ {n}  ( i.o.) \}  =  1
 +
$$
  
for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057740/l05774014.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057740/l05774015.png" /> denotes infinitely often.
+
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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057740/l05774016.png" /> is a sequence of independent random variables having identical distributions with mathematical expectations equal to zero, then
+
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
  
<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/l/l057/l057740/l05774017.png" /></td> </tr></table>
+
$$
  
(Kolmogorov's theorem); if the additional condition <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057740/l05774018.png" /> is satisfied, then one has the stronger relation (2), in which
+
\frac{S _ {n} }{n}
 +
\rightarrow 0 \  \textrm{ (a.s.) } \  \textrm{ for }  n \rightarrow \infty
 +
$$
  
<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/l/l057/l057740/l05774019.png" /></td> </tr></table>
+
(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
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057740/l05774020.png" /> (the Hartman–Wintner theorem).
+
$$
 +
c _ {n}  = ( 2nb  { \mathop{\rm ln}  \mathop{\rm ln} } ( nb))  ^ {1/2} ,
 +
$$
  
The first theorem of general type on the law of the iterated logarithm was the following result obtained by A.N. Kolmogorov [[#References|[1]]]. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057740/l05774021.png" /> be a sequence of independent random variables with mathematical expectations equal to zero and with finite variances, and let
+
where  $  b = {\mathsf E} X _ {1}  ^ {2} $(
 +
the Hartman–Wintner theorem).
  
<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/l/l057/l057740/l05774022.png" /></td> </tr></table>
+
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
  
If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057740/l05774023.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057740/l05774024.png" /> and if there exists a sequence of positive constants <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057740/l05774025.png" /> such that
+
$$
 +
B _ {n}  = \sum_{k=1}^ { n }  {\mathsf E} X _ {k}  ^ {2} .
 +
$$
  
<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/l/l057/l057740/l05774026.png" /></td> </tr></table>
+
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
 
then (1) and (2) are satisfied for
  
<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/l/l057/l057740/l05774027.png" /></td> </tr></table>
+
$$
 +
c _ {n}  = ( 2B _ {n}  \mathop{\rm ln}  \mathop{\rm ln}  B _ {n} )  ^ {1/2} .
 +
$$
  
In the particular case where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057740/l05774028.png" /> is a sequence of independent random variables having identical distributions with two possible values, this assertion was derived by A.Ya. Khinchin [[#References|[2]]]. J. Marcinkiewicz and A. Zygmund [[#References|[3]]] showed that under the conditions of Kolmogorov's theorem one cannot replace <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057740/l05774029.png" /> by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057740/l05774030.png" />. W. Feller [[#References|[4]]] examined a generalization of Kolmogorov's law of the iterated logarithm for sequences of independent bounded non-identically distributed random variables. See [[#References|[5]]] for other generalizations of the law; there is also the following result (see [[#References|[6]]]), which is related to the Hartman–Wintner theorem: If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057740/l05774031.png" /> is a sequence of independent random variables having identical distributions with infinite variances, then
+
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
  
<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/l/l057/l057740/l05774032.png" /></td> </tr></table>
+
$$
 +
\lim\limits _ {n \rightarrow \infty }  \sup 
 +
\frac{| S _ {n} | }{( n  \mathop{\rm ln}  n  \mathop{\rm ln}  n)  ^ {1/2} }
 +
  = \infty \  ( a.s.).
 +
$$
  
 
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.
 
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====
 
====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  A.N. [A.N. Kolmogorov] Kolmogoroff,   "Ueber das Gesetz des iterierten Logarithmus" ''Math. Ann.'' , '''101''' (1929) pp. 126–135</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"A. [A.Ya. Khinchin] Khintchine,   "Ueber einen Satz der Wahrscheinlichkeitsrechnung" ''Fund. Math.'' , '''6''' (1924) pp. 9–20</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  J. Marcinkiewicz,   A. Zygmund,   "Rémarque sur la loi du logarithme itéré" ''Fund. Math.'' , '''29''' (1937) pp. 215–222</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top"W. Feller,   "The general form of the so-called law of the iterated logarithm" ''Trans. Amer. Math. Soc.'' , '''54''' (1943) pp. 373–402</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top"V. Strassen,   "An invariance principle for the law of the iterated logarithm" ''Z. Wahrsch. Verw. Geb.'' , '''3''' (1964) pp. 211–226</TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top"V. Strassen,   "A converse to the law of iterated logarithm" ''Z. Wahrsch. Verw. Geb.'' , '''4''' (1965–1966) pp. 265–268</TD></TR><TR><TD valign="top">[7]</TD> <TD valign="top"P. Hartman,   A. Wintner,   "On the law of the iterated logarithm" ''Amer. J. Math.'' , '''63''' (1941) pp. 169–176</TD></TR><TR><TD valign="top">[8]</TD> <TD valign="top"J. Lamperty,   "Probability" , Benjamin (1966)</TD></TR><TR><TD valign="top">[9]</TD> <TD valign="top">  V.V. Petrov,   "Sums of independent random variables" , Springer (1975) (Translated from Russian)</TD></TR></table>
+
{|
 
+
|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}}
====Comments====
+
|-
 
+
|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}}
 
+
|-
====References====
+
|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}}
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  P. Hall,   C.C. Heyde,   "Martingale limit theory and its application" , Acad. Press (1980)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  W. Feller,   "An introduction to probability theory and its applications" , '''1''' , Wiley (1968)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  M. Loève,   "Probability theory" , Princeton Univ. Press (1963) pp. Chapt. XIV</TD></TR></table>
+
|-
 +
|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}}
 +
|}

Latest revision as of 08:56, 21 January 2024


2020 Mathematics Subject Classification: Primary: 60F10 Secondary: 60F15 [MSN][ZBL]

A limit theorem in probability theory which is a refinement of the 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 [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 [Kh]. J. Marcinkiewicz and A. Zygmund [MZ] showed that under the conditions of Kolmogorov's theorem one cannot replace $ o $ by $ O $. W. Feller [F] examined a generalization of Kolmogorov's law of the iterated logarithm for sequences of independent bounded non-identically distributed random variables. See [S] for other generalizations of the law; there is also the following result (see [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

[Ko] A.N. Kolmogoroff, "Ueber das Gesetz des iterierten Logarithmus" Math. Ann. , 101 (1929) pp. 126–135
[Kh] A. Khintchine, "Ueber einen Satz der Wahrscheinlichkeitsrechnung" Fund. Math. , 6 (1924) pp. 9–20 Zbl 50.0344.02
[MZ] J. Marcinkiewicz, A. Zygmund, "Rémarque sur la loi du logarithme itéré" Fund. Math. , 29 (1937) pp. 215–222 Zbl 0018.03204 Zbl 63.1076.03
[F] W. Feller, "The general form of the so-called law of the iterated logarithm" Trans. Amer. Math. Soc. , 54 (1943) pp. 373–402 MR0009263 Zbl 0063.08417
[S] V. Strassen, "An invariance principle for the law of the iterated logarithm" Z. Wahrsch. Verw. Geb. , 3 (1964) pp. 211–226 MR0175194 Zbl 0132.12903
[S2] V. Strassen, "A converse to the law of iterated logarithm" Z. Wahrsch. Verw. Geb. , 4 (1965–1966) pp. 265–268 Zbl 0141.16501
[HW] P. Hartman, A. Wintner, "On the law of the iterated logarithm" Amer. J. Math. , 63 (1941) pp. 169–176 MR0003497 Zbl 0024.15802 Zbl 67.0460.03
[La] J. Lamperty, "Probability" , Benjamin (1966)
[P] V.V. Petrov, "Sums of independent random variables" , Springer (1975) (Translated from Russian) MR0388499 Zbl 0322.60043 Zbl 0322.60042
[HH] P. Hall, C.C. Heyde, "Martingale limit theory and its application" , Acad. Press (1980) MR0624435 Zbl 0462.60045
[F2] W. Feller, "An introduction to probability theory and its applications" , 1 , Wiley (1968) MR0228020 Zbl 0155.23101
[Lo] M. Loève, "Probability theory" , Princeton Univ. Press (1963) pp. Chapt. XIV MR0203748 Zbl 0108.14202
How to Cite This Entry:
Law of the iterated logarithm. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Law_of_the_iterated_logarithm&oldid=11246
This article was adapted from an original article by V.V. Petrov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article