Infinite divisibility (probability)

The concept of infinite divisibility (also known as unrestricted or unlimited divisibility ) describes the stochastic property of many random variables, can be broken down as the sum of individual independent random variables. The term was introduced in 1929 by the Italian-Austrian mathematician Bruno de Finetti. It is closely related to the concept of reproductivity (but not identical, see below) and plays a major role, especially in the theory of Lévy processes.

Definition

Let be a probability space and a d-dimensional random variable on. X is called infinitely divisible on this probability space, if there is for every random variable with

  • Are independent and identically distributed
  • .

Particularly great importance to the concept of infinite divisibility in the following two sub- areas of stochastics to:

Infinite divisibility and sums of independent random variables

In the general summation theory of independent random variables is considered consequences X1, X2, ..., Xn, ... of random variables, each of which is a sum of finitely many independent and identically distributed random variables Xn1, xn2, ..., Xn kn. Then the following statement holds:

If none of the individual summands XNK has a significant impact on the sum of ( mathematically formulated as a condition of "infinite smallness " for each one), then the standardized distribution functions Fn converge to an infinitely divisible distribution function F.

In other words, the class of infinitely divisible distribution functions is identical with the class of limit distributions for sums of independent and identically distributed random variables. These statements are due to Kolmogorov and his student Khinchin and Gnedenko.

Infinite divisibility and Lévy processes

For random variables A and B exists if a Lévy process with states, if the random variable BA is infinitely divisible. This result of Paul Lévy simplifies the proof of the existence of Brownian motion (first proved by Norbert Wiener in 1923 ) dramatically, as can easily be shown that the normal distribution is infinitely divisible.

Examples

  • As already mentioned, each normally distributed random variable is infinitely divisible: for selected independent. So that the above conditions are satisfied.
  • The exponential distribution with expected value is infinitely divisible, the corresponding " dividers " are gamma distributed with mean and variance. ( Note the non-uniform parameterization ).
  • There are also infinitely divisible discrete random variable: For the Poisson distribution with parameter is infinitely divisible: here the independent summands X1, X2, ..., Xn are also Poisson distributed with parameter.
  • Other examples of infinitely divisible random variable is the gamma distribution (to the chi-square distribution and the exponential distribution ), the lognormal distribution, the logistic distribution, the Pareto - distribution Dirac distribution, the negative binomial distribution, alpha- stable distribution, the Gumbel distribution, the F distribution and the student's distribution, as well as the inverse gaussian and the normal inverse gaussian distributions.
  • One can see immediately that the Bernoulli distribution, characterized by P (X = 1 ) = p and P ( X = 0) = 1 - p, is not infinitely divisible: for this purpose are X1 and X2 are independent for n = 2, the same distributed summands X1 X2 = X. If this would be trivial ( ie, if they could take only one value ), the sum would also be trivial. So have X1 and X2 at least two different values ​​with positive probability assume about. The sum X1 X2 would then each with positive probability to accept the three pairwise different values ​​2a, 2b and a b and would therefore not Bernoulli distributed. So can not exist X1 and X2. Similarly it can be shown that a non-trivial distribution that takes only finitely many values ​​that can not be infinitely divisible.
  • With a little more effort can be shown that the continuous uniform distribution is also not infinitely divisible.

Alternative definitions and canonical representations

In the above definition of the random variables were assumed by the concept. They can be applied to distribution functions taking into account that the distribution function of a sum of independent and identically distributed random variables is the convolution of the distribution functions of the summands:

A distribution function F is infinitely divisible if and only if there exists a distribution function for each n > 0, such that where n * denotes the n- fold convolution.

Considering nor the associated characteristic functions, noting that the characteristic function of a convolution is the product of the characteristic functions of the convolution factors, then you get another equivalent definition of infinite divisibility:

A characteristic function f is infinitely divisible if there exists a characteristic function for each n > 0 such that

In particular, by this very simple definition can be in some cases the question of infinite divisibility easily answered. So, for example, has The above example Chi -square distribution with parameter m, the characteristic function f (t) = 1 / ( 1 - 2IT ) m / 2 and fn (t) = 1 / (1 - 2IT ) (m / n ) / 2 is again a characteristic function of a chi-square distribution with parameter m / n

From the last definition, canonical representations of infinitely divisible distribution functions can be derived: A distribution function F ( x) is infinitely divisible if and only if its characteristic function f ( t) one of the following representations has

( canonical representation by Paul Lévy and Khinchin Alexandr ) or

( canonical representation according to Lévy ).

Where A and real numbers, M is a monotonically non-decreasing function and limited, and M and N are not monotonically decreasing and the integrals with and and exist for each.

Both representations are unique.

The parameter a is just one horizontal displacement of the distribution function F on the real axis (offset parameter, engl. "Location parameter "). The constant is called the Gaussian component. The function H is called Lévy- Chintschinsche spectral function of F or f, it has up to a non-negative factor the properties of a distribution function, the functions M and N are Lévysche spectral functions of F or f

These two canonical representations are generalizations of previously discovered by Andrei Kolmogorov representation, which only applies to distribution functions with existing variance.

Infinite divisibility vs.. reproductivity

A similar attribute for random variables is the reproductivity, which states that a family of distributions is reproductively, if the sum of two independent random variables from the family is back in the same family. A difference to the infinite divisibility is, for example, that in the latter, the family must not be specified:

So the family of exponential distributions is infinitely divisible, but not reproductively ( the exponential distributions, however, form a subfamily of gamma distributions, which in turn are reproductively ).

An example of a reproductive, but not infinitely divisible family on the other hand, the binomial distribution n variable parameters and fixed parameters p: For example, if X Binomial ( n, p) distributed and Y independent of Binomial (m, p) - distributed, as has X Y is a binomial (m n, p) distribution. Infinitely divisible X is not, since it can not be decomposed into n 1 identical independent summand for example.

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