View source for Law of the iterated logarithm

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

<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>

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

<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>

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>

Relation (1) is equivalent to

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and

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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.

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

<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

<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>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057740/l05774020.png" /> (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 [[#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

<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>

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

<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>

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>

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

<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>

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====
<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>



====Comments====


====References====
<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>