Statistical methods related to the law of the iterated logarithm

  • What is the law of iterated logarithm in statistics?

    The law of the iterated logarithm says that if Xn is a sequence of iid random variables with zero expectation and unit variance, then the partial sums sequence Sn=∑ni=.

    1. Xi satisfies almost surely that lim supn→∞Sn√2nloglog n=1
      2. .Dec 26, 2015

  • Iterated Logarithm or Log*(n) is the number of times the logarithm function must be iteratively applied before the result is less than or equal to 1.
October, 1970 Statistical Methods Related to the Law of the Iterated Logarithm. Herbert RobbinsDOWNLOAD PDF + SAVE TO MY LIBRARY. Ann. Math. Statist.
We shall give a method for obtaining probability inequalities and related limit theorems concerning the behavior of the entire sequence of x's. We begin with a 

What are the limit theorems for sums of independent random variables?

For sums of independent random variables we already know two limit theorems:

  1. the law of large numbers and the central limit theorem

The law of large numbers describes for large (nin {mathbb {N}}), the typical behavior, or average value behavior, of sums of n random variables.
,

What does Slivka's mean in a logarithm?

For proofs and related material we refer to [61, 62, 105].
The law of the iterated logarithm tells us that L(ε) < almost ∞ ε > σ√2.
Slivka’s result implies that no moments exist.
Relation (7.8) that logarithmic moments exist, but not all the way down to σ√2, only However, in it is shown that if Var X = σ2 < , then E log log ∞ for ε > σ√2.

,

What is the law of the iterated logarithm?

In probability theory, the law of the iterated logarithm describes the magnitude of the fluctuations of a random walk.
The original statement of the law of the iterated logarithm is due to A.
Ya.
Khinchin (1924). Another statement was given by A.
N.
Kolmogorov in 1929.

,

Which law of the logarithm is the hottest candidate?

The answer to these questions belong to the realm of the law of the logarithm.
If the law of large numbers and the central limit theorem two most central and fundamental limit theorems, the law of the logarithm is the hottest candidate for the third position.

In mathematics, the incompressibility method is a proof method like the probabilistic method, the counting method or the pigeonhole principle.
To prove that an object in a certain class satisfies a certain property, select an object of that class that is incompressible.
If it does not satisfy the property, it can be compressed by computable coding.
Since it can be generally proven that almost all objects in a given class are incompressible, the argument demonstrates that almost all objects in the class have the property involved.
To select an incompressible object is ineffective, and cannot be done by a computer program.
However, a simple counting argument usually shows that almost all objects of a given class can be compressed by only a few bits.
This is a list of probability topics.
It overlaps with the (alphabetical) list of statistical topics.
There are also the outline of probability and catalog of articles in probability theory.
For distributions, see List of probability distributions.
For journals, see list of probability journals.
For contributors to the field, see list of mathematical probabilists and list of statisticians.

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