Law of the iterated logarithm

As a law of the iterated logarithm several limit theorems of probability theory are known. You make statements about the asymptotic behavior of sums of random variables or 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 mean 0 and variance 1 Then

And

The law of the iterated logarithm completes an important statement about the asymptotic behavior of sums of random variables, the law of large numbers and the central limit theorem. First evidence in simple cases submitted by Khinchin (1924 ) and Kolmogorov (1929 ), the proof given here for the general case in 1941 provided by Hartman and Wintner.

The laws of the iterated logarithm for the Wiener process

In the following it is always a standard Wiener process on a suitable probability space, ie each is represented by a function. The course of this function is dependent, so random. Moreover, the variance, that is, the measure of the "uncertainty " of W grows, t increases to infinity. All the more amazing it seems that it is possible with the help of the laws of the iterated logarithm meet so precise statements about the Wiener process:

The first law

The first law of the iterated logarithm says:

It called limsup the limes superior and loglog is the twice running ( iterated ) natural logarithm.

The law can be interpreted as follows: Looking for an arbitrarily small, the two functions

So there is always a ( by and dependent ) time, so

• ( Ie is never exceeded more)
• ( Ie is still being exceeded ).

The second law

The second law of iterated logarithm treated the limes inferior of the Wiener process and is a simple consequence of the first: as is true for all time points ( here denotes the normal distribution ) and W is therefore particularly symmetrically distributed around zero, it follows

The interpretation of this fact also recognized completely analogous: one replaces the features and simply by their negative and the verb "to exceed " to " below ." Noteworthy here is the particular combination of the two laws: while the outer limits and not at some point be reached, the inner, only marginally distant from the outer limits of boundaries and yet both are infinitely often crossed. The Wiener process must therefore always between the two boundaries and heroszillieren and, in particular, infinitely change the sign.

The third and fourth law

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

Analogous to the above interpretation is considered to be any two functions

Then there is again a (this time possibly very small ), so that

• Always applies to everyone, but
• It is all still there with
• But also for all still there with.

Since here both laws apply at the same time, it means that the Wiener process almost certainly in every small interval infinitely often changes sign and (since the Wiener process is almost surely continuous and thus the intermediate value theorem is sufficient) there infinitely many zeros has.

To prove the laws

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

Produced, which converts the two problems together. It remains only to prove the first law. This evidence first was made in 1929 the Russian mathematician Alexander Khinchin, ie six years after Norbert Wiener had proved the existence of the Wiener process. It was later followed by yet another, far more elegant proof by Paul Lévy using the martingale theory, which was not yet known Khinchin.