Law of the iterated logarithm
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 be a sequence of random variables and let
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For simplicity one assumes that has zero median for each
. While the theorem on the strong law of large numbers deals with conditions under which
almost surely (
) for
, where
is a sequence of numbers, the theorem on the law of the iterated logarithm deals with sequences of numbers
such that
![]() | (1) |
or
![]() | (2) |
Relation (1) is equivalent to
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and
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for any , where
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 is a sequence of independent random variables having identical distributions with mathematical expectations equal to zero, then
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(Kolmogorov's theorem); if the additional condition is satisfied, then one has the stronger relation (2), in which
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where (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 [1]. Let be a sequence of independent random variables with mathematical expectations equal to zero and with finite variances, and let
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If for
and if there exists a sequence of positive constants
such that
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then (1) and (2) are satisfied for
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In the particular case where is a sequence of independent random variables having identical distributions with two possible values, this assertion was derived by A.Ya. Khinchin [2]. J. Marcinkiewicz and A. Zygmund [3] showed that under the conditions of Kolmogorov's theorem one cannot replace
by
. W. Feller [4] examined a generalization of Kolmogorov's law of the iterated logarithm for sequences of independent bounded non-identically distributed random variables. See [5] for other generalizations of the law; there is also the following result (see [6]), which is related to the Hartman–Wintner theorem: If
is a sequence of independent random variables having identical distributions with infinite variances, then
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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
[1] | A.N. [A.N. Kolmogorov] Kolmogoroff, "Ueber das Gesetz des iterierten Logarithmus" Math. Ann. , 101 (1929) pp. 126–135 |
[2] | A. [A.Ya. Khinchin] Khintchine, "Ueber einen Satz der Wahrscheinlichkeitsrechnung" Fund. Math. , 6 (1924) pp. 9–20 Zbl 50.0344.02 |
[3] | 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 |
[4] | 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 |
[5] | 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 |
[6] | V. Strassen, "A converse to the law of iterated logarithm" Z. Wahrsch. Verw. Geb. , 4 (1965–1966) pp. 265–268 Zbl 0141.16501 |
[7] | 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 |
[8] | J. Lamperty, "Probability" , Benjamin (1966) |
[9] | V.V. Petrov, "Sums of independent random variables" , Springer (1975) (Translated from Russian) MR0388499 Zbl 0322.60043 Zbl 0322.60042 |
Comments
References
[a1] | P. Hall, C.C. Heyde, "Martingale limit theory and its application" , Acad. Press (1980) MR0624435 Zbl 0462.60045 |
[a2] | W. Feller, "An introduction to probability theory and its applications" , 1 , Wiley (1968) MR0228020 Zbl 0155.23101 |
[a3] | M. Loève, "Probability theory" , Princeton Univ. Press (1963) pp. Chapt. XIV MR0203748 Zbl 0108.14202 |
Law of the iterated logarithm. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Law_of_the_iterated_logarithm&oldid=21487