Law of the iterated logarithm

As a law of the iterated logarithm several are limit theorems of probability theory called. They make statements about the asymptotic behavior of sums of random variables or of stochastic processes .

The law of the iterated logarithm for sums of random variables

Let be a sequence of independent, identically distributed (iid) random variables with expected value 0 and variance 1. Then applies ${\displaystyle X_{1},X_{2},\dots }$

${\displaystyle \limsup _{N\to \infty }{\frac {\sum _{k=1}^{N}X_{k}}{\sqrt {2N\log \log N}}}=1}$       pretty sure

and

${\displaystyle \liminf _{N\to \infty }{\frac {\sum _{k=1}^{N}X_{k}}{\sqrt {2N\log \log N}}}=-1}$       almost sure .

The law of the iterated logarithm, as an important statement about the asymptotic behavior of sums of random variables, completes the law of large numbers and the central limit theorem . The first evidence in simple cases comes from Chintschin (1924) and Kolmogorow (1929), the evidence for the general case cited here was provided in 1941 by Philip Hartman and Aurel Wintner . This is why the statement is also referred to as the Hartman-Wintner theorem . A proof is possible, for example, using the Skorochod embedding theorem in combination with the statement for the Wiener process.

The laws of the iterated logarithm for the Wiener process

In the following, a standard Wiener process is always assumed on a suitable probability space , i.e. H. for each is given by a function . The course of this function is dependent, i.e. random. In addition, the variance , ie the measure of the "indeterminacy" of W, grows to infinity with increasing t. It seems all the more astonishing that such precise statements about the Wiener process can be made with the help of the laws of the iterated logarithm: ${\displaystyle (W_{t}),\;t\geq 0}$ ${\displaystyle (\Omega ,{\mathcal {F}},P)}$ ${\displaystyle \omega \in \Omega }$ ${\displaystyle W_{t}(\omega )}$ ${\displaystyle [0,\infty ]\to \mathbb {R} }$ ${\displaystyle \omega }$

The first law

The first two laws are shown graphically: the picture shows four independent Wiener processes with 0 <t <1000 and the asymptotic envelopes

The first law of the iterated logarithm says:

${\displaystyle \limsup _{t\to \infty }{\frac {W_{t}(\omega )}{\sqrt {2t\log \log(t)}}}=1}$for P-almost-everyone . ${\displaystyle \omega \in \Omega }$

Here referred limsup the limes superior and loglog of twice is carried out (iterated) natural logarithm .

The law can be interpreted as follows: If one considers the two functions for an arbitrarily small one ${\displaystyle \epsilon \in \mathbb {R} ,\;\epsilon >0}$

${\displaystyle f_{+}(x)=(1+\epsilon ){\sqrt {2x\log \log(x)}}}$ and

${\displaystyle f_{-}(x)=(1-\epsilon ){\sqrt {2x\log \log(x)}}}$,

so there is always a (of and dependent) point in time , so that ${\displaystyle \epsilon }$ ${\displaystyle \omega }$ ${\displaystyle 0<T_{\epsilon }(\omega )<\infty }$

• ${\displaystyle \forall \;t\geq T_{\epsilon }(\omega ):\;W_{t}(\omega )<f_{+}(t)}$ ( will never be exceeded again) ${\displaystyle f_{+}}$
• ${\displaystyle \forall \;T\geq T_{\epsilon }(\omega ):\;\exists \;t\geq T:\;W_{t}(\omega )>f_{-}(t)}$( is therefore always exceeded). ${\displaystyle f_{-}}$

The second law

The second law of the iterated logarithm deals with the limes inferior of the Wiener process and is a simple consequence of the first: since it applies to all points in time ( denotes the normal distribution ) and W is therefore particularly symmetrically distributed around the zero point, it follows from this ${\displaystyle t\geq 0\;\;W_{t}\sim {\mathcal {N}}(0,t)}$ ${\displaystyle {\mathcal {N}}}$

${\displaystyle \liminf _{t\to \infty }{\frac {W_{t}(\omega )}{\sqrt {2t\log \log(t)}}}=-1}$for P-almost-everyone . ${\displaystyle \omega \in \Omega }$

The interpretation of this fact is also completely analogous: you replace the functions and simply with their negative and the verb "exceed" with "fall below". The combination of the two laws is particularly noteworthy: while the outer limits and at some point are no longer reached, the inner limits, which are only marginally distant from the outer limits, and both are crossed infinitely often. The Viennese trial has to oscillate back and forth between the two borders again and again and, in particular, change its sign infinitely often . ${\displaystyle f_{+}}$ ${\displaystyle f_{+}}$ ${\displaystyle f_{+}}$ ${\displaystyle -f_{+}}$ ${\displaystyle f_{-}}$ ${\displaystyle -f_{-}}$

The third and fourth law

The corresponding graph for laws three and four. Here the time horizon is 0 <t <0.01

The other two laws of the iterated logarithm are less clear than the first two, since they describe the behavior of the Wiener process not in an unlimited, but only in a very small interval, namely around the zero point. The following applies there:

${\displaystyle \limsup _{t\to 0}{\frac {W_{t}(\omega )}{\sqrt {2t\log \log(1/t)}}}=1}$ such as

${\displaystyle \liminf _{t\to 0}{\frac {W_{t}(\omega )}{\sqrt {2t\log \log(1/t)}}}=-1}$each for P-almost-all . ${\displaystyle \omega \in \Omega }$

Analogous to the above interpretation, one considers the two functions for any one ${\displaystyle \epsilon >0}$

${\displaystyle g_{+}(x)=(1+\epsilon ){\sqrt {2x\log \log {\frac {1}{x}}}}}$ and

${\displaystyle g_{-}(x)=(1-\epsilon ){\sqrt {2x\log \log {\frac {1}{x}}}}}$.

Then there is another one (this time possibly very small) so that ${\displaystyle T_{\epsilon }(\omega )>0}$

• For everyone always applies, but ${\displaystyle 0<t\leq T_{\epsilon }(\omega )}$ ${\displaystyle -g_{+}(t)<W_{t}(\omega )<g_{+}(t)}$
• It all still there with ${\displaystyle 0<T\leq T_{\epsilon }(\omega )}$ ${\displaystyle t\in {]0,T[}}$ ${\displaystyle W_{t}(\omega )>g_{-}(t)}$
• it but also for all still there with . ${\displaystyle 0<T\leq T_{\epsilon }(\omega )}$ ${\displaystyle t\in {]0,T[}}$ ${\displaystyle W_{t}(\omega )<-g_{-}(t)}$

Since both laws also apply here at the same time, this means that the Wiener process almost certainly changes its sign infinitely often in every interval, no matter how small , and (since the Wiener process is almost certainly continuous and thus satisfies the intermediate value theorem) there an infinite number of zeros Has. ${\displaystyle ]0,t[}$

To prove the law

As already mentioned, laws 1 and 2 are equivalent due to the symmetry of the normal distribution, which also applies to laws 3 and 4. Furthermore, an equivalence between the first and the third law can quickly be found on the basis of self-similarity

${\displaystyle W_{t}\sim tW_{\frac {1}{t}}}$

produce that merges the two problems. So it only remains to prove the first law. The Russian mathematician Alexandr Chintschin was first able to prove this in 1929 , six years after Norbert Wiener had proven the existence of the Wiener process. Another, far more elegant proof followed later by Paul Lévy using the martingale theory , which Chintschin was not yet aware of.

literature

• Achim Klenke: Probability Theory. 2nd Edition. Springer-Verlag, Berlin Heidelberg 2008, ISBN 978-3-540-76317-8 , chap. 22nd