# 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.

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=47596
This article was adapted from an original article by V.V. Petrov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article