Law of the iterated logarithm

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

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

and

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

(Kolmogorov's theorem); if the additional condition is satisfied, then one has the stronger relation (2), in which

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

If for and if there exists a sequence of positive constants such that

then (1) and (2) are satisfied for

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

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
[3]	J. Marcinkiewicz, A. Zygmund, "Rémarque sur la loi du logarithme itéré" Fund. Math. , 29 (1937) pp. 215–222
[4]	W. Feller, "The general form of the so-called law of the iterated logarithm" Trans. Amer. Math. Soc. , 54 (1943) pp. 373–402
[5]	V. Strassen, "An invariance principle for the law of the iterated logarithm" Z. Wahrsch. Verw. Geb. , 3 (1964) pp. 211–226
[6]	V. Strassen, "A converse to the law of iterated logarithm" Z. Wahrsch. Verw. Geb. , 4 (1965–1966) pp. 265–268
[7]	P. Hartman, A. Wintner, "On the law of the iterated logarithm" Amer. J. Math. , 63 (1941) pp. 169–176
[8]	J. Lamperty, "Probability" , Benjamin (1966)
[9]	V.V. Petrov, "Sums of independent random variables" , Springer (1975) (Translated from Russian)

References

[a1]	P. Hall, C.C. Heyde, "Martingale limit theory and its application" , Acad. Press (1980)
[a2]	W. Feller, "An introduction to probability theory and its applications" , 1 , Wiley (1968)
[a3]	M. Loève, "Probability theory" , Princeton Univ. Press (1963) pp. Chapt. XIV