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 $ 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