iterated logarithm

The question then becomes: on which level of iterated logarithms do we wish to compare two numbers?

The iterated logarithm accepts any positive real number and yields an integer.

Such comparisons of iterated logarithms are common, e.g., in analytic number theory.

On the law of the iterated logarithm.

The law of the iterated logarithm provides the scaling factor where the two limits become different:

The iterated logarithm grows at an extremely slow rate, much slower than the logarithm itself.

Higher bases give smaller iterated logarithms.

The law of iterated logarithms operates "in between" the law of large numbers and the central limit theorem.

In probability theory, the law of the iterated logarithm describes the magnitude of the fluctuations of a random walk.

The generalized logarithm function is closely related to the iterated logarithm used in computer science analysis of algorithms.

The central limit theorem and the law of the iterated logarithm describe important aspects of the behavior of simple random walk on .

Feller made fundamental contributions to renewal theory, Tauberian theorems, random walks, diffusion processes, and the law of the iterated logarithm.

Mathematically, the iterated logarithm is well-defined not only for base 2 and base e, but for any base greater than .

An even stronger uniform convergence result for the empirical distribution function is available in the form of an extended type of law of the iterated logarithm.

His thesis was in the area of probability in Banach spaces, and solved a problem related to the law of the iterated logarithm for empirical characteristic functions.

S. Ralescu, [6], The law of the iterated logarithm for the multivariate nearest neighbor density estimators, J. Multivar.

The iterated logarithm is useful in analysis of algorithms and computational complexity, appearing in the time and space complexity bounds of some algorithms such as:

Statement: If m operations, either Union or Find, are applied to n elements, the total run time is O(m logn), where log is the iterated logarithm.

For any fixed base, the sum of the digits of a number is proportional to its logarithm; therefore, the additive persistence is proportional to the iterated logarithm.

This result, now known as Strassen's invariance principle or as Strassen's law of the iterated logarithm, has been highly cited and led to a 1966 presentation at the International Congress of Mathematicians.

More strongly, I. J. Good suggests that the convergence rate of this limit should be significantly faster than that of a random binary sequence, for which (by the law of the iterated logarithm) .

For all values of n relevant to counting the running times of algorithms implemented in practice (i.e., n 2, which is far more than the atoms in the known universe), the iterated logarithm with base 2 has a value no more than 5.

He became one of the founders of modern probability theory, discovering the law of the iterated logarithm in 1924, achieving important results in the field of limit theorems, giving a definition of a stationary process and laying a foundation for the theory of such processes.

Strassen began his researches as a probabilist; his 1964 paper An Invariance Principle for the Law of the Iterated Logarithm defined a functional form of the law of the iterated logarithm, showing a form of scale invariance in random walks.