Infinite divisibility

The term infinite divisibility (also referred to as unlimited or unlimited divisibility ) describes in stochastics the property of many random variables that 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 in the theory of the Lévy processes .

definition

Let be a probability space and a -dimensional random variable on it. is called infinitely divisible on this probability space if there are random variables with for each ${\displaystyle (\Omega ,{\mathcal {A}},P)}$ ${\displaystyle X\colon \Omega \to \mathbb {R} ^{d}}$ ${\displaystyle d}$ ${\displaystyle X}$ ${\displaystyle n\in \mathbb {N} }$ ${\displaystyle X_{1},X_{2},\dotsc ,X_{n}\colon \Omega \to \mathbb {R} ^{d}}$

• ${\displaystyle X_{1},X_{2},\dotsc ,X_{n}}$are distributed independently and identically
• ${\displaystyle \sum _{i=1}^{n}X_{i}\sim X}$.

The concept of infinite divisibility is particularly important in the following two sub-areas of stochastics:

Infinite divisibility and sums of independent random variables

In the general summation theory for independent random variables one considers sequences of random variables, each of which is a sum of finitely many independent and identically distributed random variables . Then the following statement applies: ${\displaystyle X_{1},X_{2},\dotsc ,X_{n},\dotsc }$ ${\displaystyle X_{n1},X_{n2},\dotsc ,X_{nk_{n}}}$

If none of the individual totals has a significant influence on the sum (formulated mathematically as a condition of "infinite smallness" for each , see also asymptotically negligible scheme ), then the standardized distribution functions converge to an infinitely divisible distribution function . ${\displaystyle X_{nk}}$ ${\displaystyle \lim _{n\to \infty }\max _{k}\;P(\left|X_{nk}\right|>\epsilon )=0\;}$ ${\displaystyle \epsilon >0}$ ${\displaystyle F_{n}}$ ${\displaystyle F}$

In other words, the class of the infinitely divisible distribution functions is identical to the class of the limit distributions for sums of independent and identically distributed random variables. These statements go back to Kolmogorow and his students Chintschin and Gnedenko.

Infinite divisibility and Lévy processes

For random variables and there is a Lévy process with states if and only if the random variable is infinitely divisible. This result by Paul Lévy dramatically simplifies the proof of the existence of Brownian motion (first proven by Norbert Wiener in 1923), since it can easily be shown that the normal distribution is infinitely divisible. ${\displaystyle A}$ ${\displaystyle B}$ ${\displaystyle (X_{t}),\;t\in \mathbb {Q} }$ ${\displaystyle X_{0}\sim A,X_{1}\sim B}$ ${\displaystyle B-A}$

Examples

• As already mentioned, every normally distributed random variable is infinitely divisible: for choose independent . The above conditions are thus met. ${\displaystyle X\sim {\mathcal {N}}(\mu ,\sigma ^{2}),\;\;n\in \mathbb {N} }$ ${\displaystyle X_{1},X_{2},\dotsc ,X_{n}\sim {\mathcal {N}}\left({\frac {\mu }{n}},{\frac {\sigma ^{2}}{n}}\right)}$

• The exponential distribution with expected value is infinitely divisible, the associated “divisors” are gamma distributed with expected value and variance . (Note the inconsistent parameterization). ${\displaystyle \tau }$ ${\displaystyle {\tfrac {1}{n}}\cdot \tau }$ ${\displaystyle {\tfrac {1}{n}}\cdot \tau ^{2}}$

• There are also discrete, infinitely divisible random variables: For example, the Poisson distribution with parameters is infinitely divisible: here the independent summands are also Poisson distributed with parameters . ${\displaystyle \lambda >0}$ ${\displaystyle X_{1},X_{2},\dotsc ,X_{n}}$ ${\displaystyle \lambda /n}$

• Further examples of infinitely divisible random variables are the gamma distribution (thus chi-square distribution and exponential distribution ), the logarithmic normal distribution , the logistic distribution , the Pareto distribution , the Dirac distribution , the negative binomial distribution , alpha-stable distributions , the Gumbel distribution , the F distribution and the Student distribution , as well as the inverse Gaussian and the normally inverse Gaussian distributions.

• One quickly sees that the Bernoulli distribution, characterized by and with, is not infinitely divisible: For let this and the independent, identically distributed summands with . If these were trivial (i.e. if they could only take one value), the sum would also be trivial. So have and assume at least two different values with positive probability, approximately . The sum would then assume the three different paired values and with a positive probability in each case and would therefore not be Bernoulli-distributed. So can and not exist. Analogously it can be shown that a nontrivial distribution that only takes on a finite number of values ​​is not infinitely divisible. ${\displaystyle P(X=1)=p}$ ${\displaystyle P(X=0)=1-p}$ ${\displaystyle 1>p>0}$ ${\displaystyle n=2}$ ${\displaystyle X_{1}}$ ${\displaystyle X_{2}}$ ${\displaystyle X_{1}+X_{2}=X}$ ${\displaystyle X_{1}}$ ${\displaystyle X_{2}}$ ${\displaystyle a,b\in \mathbb {R} \;\;(a\neq b)}$ ${\displaystyle X_{1}+X_{2}=X}$ ${\displaystyle 2a,2b}$ ${\displaystyle a+b}$ ${\displaystyle X_{1}}$ ${\displaystyle X_{2}}$

• With a little more effort it can be shown that the constant uniform distribution is also not infinitely divisible.

Alternative definitions and canonical representations

In the definition above, the concept of random variables was assumed. It can be transferred to distribution functions if one takes 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 is then exactly infinitely divisible, if for each distribution function exist, so that , with the fold folding means. ${\displaystyle F}$ ${\displaystyle n>0}$ ${\displaystyle F_{n}}$ ${\displaystyle F={F_{n}}^{n*}}$ ${\displaystyle n*}$ ${\displaystyle n}$

If one also considers the associated characteristic functions and observes that the characteristic function of a convolution is the product of the characteristic functions of the convolution factors, then one obtains another equivalent definition for infinite divisibility:

A characteristic function is infinitely divisible if and only if there is a characteristic function for each such that . ${\displaystyle f}$ ${\displaystyle n>0}$ ${\displaystyle f_{n}}$ ${\displaystyle f={f_{n}}^{n}}$

This very simple definition in particular makes it easy to answer the question of infinite divisibility in some cases. So has z. B. the chi-square distribution with parameters cited above as an example the characteristic function and it is again a characteristic function of a chi-square distribution with parameters . ${\displaystyle m}$ ${\displaystyle f(t)={\frac {1}{(1-2it)^{\frac {m}{2}}}}}$ ${\displaystyle f_{n}(t)={\frac {1}{(1-2it)^{\frac {m}{2n}}}}}$ ${\displaystyle {\tfrac {m}{n}}}$

Canonical representations for infinitely divisible distribution functions can be derived from the last definition: A distribution function is infinitely divisible if and only if its characteristic function has one of the following representations ${\displaystyle F(x)}$ ${\displaystyle f(t)}$

${\displaystyle \log f(t)=iat+\int \limits _{-\infty }^{\infty }\left(e^{itu}-1-{\frac {itu}{1+u^{2}}}\right){\frac {1+u^{2}}{u^{2}}}\,dH(u)}$

( Lévy-Khinchin formula according to Paul Lévy and Alexandr Chintschin ) or

${\displaystyle \log f(t)=iat-{\frac {\sigma ^{2}}{2}}\cdot t^{2}+\int \limits _{\infty }^{-0}\left(e^{itu}-1-{\frac {itu}{1+u^{2}}}\right)\,dM(u)+\int \limits _{+0}^{\infty }\left(e^{itu}-1-{\frac {itu}{1+u^{2}}}\right)\,dN(u)}$

(canonical representation after Lévy).

Where and are real numbers, is a monotonically non-decreasing and bounded function with and and are in or monotonically non-decreasing with and the integrals and exist for each . ${\displaystyle a}$ ${\displaystyle \sigma }$ ${\displaystyle H}$ ${\displaystyle H(-\infty )=0}$ ${\displaystyle M}$ ${\displaystyle N}$ ${\displaystyle (-\infty ,0)}$ ${\displaystyle (0,\infty )}$ ${\displaystyle M(-\infty )=N(\infty )=0}$ ${\displaystyle \int \limits _{-\epsilon }^{-0}u^{2}\,dM(u)}$ ${\displaystyle \int \limits _{+0}^{\epsilon }u^{2}\,dN(u)}$ ${\displaystyle \epsilon >0}$

Both representations are clear.

The parameter only indicates a horizontal shift of the distribution function on the real axis (shift parameter, "location parameter"). The constant is called the Gaussian component. The function is called Lévy-Chintschin spectral function of or , apart from a non-negative factor it has the properties of a distribution function, the functions and are called Lévy spectral functions of or . ${\displaystyle a}$ ${\displaystyle F}$ ${\displaystyle \sigma }$ ${\displaystyle H}$ ${\displaystyle F}$ ${\displaystyle f}$ ${\displaystyle M}$ ${\displaystyle N}$ ${\displaystyle F}$ ${\displaystyle f}$

These two canonical representations are generalizations of a representation found earlier by Andrei Kolmogorow , which, however, only applies to distribution functions with existing variance .

Infinite divisibility vs. Reproductivity

A similar attribute for random variables is reproductive : a family of distributions is called reproductive if the distribution of the sum of two independent random variables with distribution from the family is again in the same family. One difference to infinite divisibility is, for example, that the family does not have to be specified for the latter:

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

An example of a reproductive, but not infinitely divisible family is the binomial distribution with variable parameters and fixed parameters : If, for example, binomial -distributed and regardless of this, binomial -distributed, then a binomial -distribution has. But it is not infinitely divisible because it cannot be broken down into identical, independent summands, for example. ${\displaystyle n}$ ${\displaystyle p}$ ${\displaystyle X}$ ${\displaystyle (n,p)}$ ${\displaystyle Y}$ ${\displaystyle (m,p)}$ ${\displaystyle X+Y}$ ${\displaystyle (m+n,p)}$ ${\displaystyle X}$ ${\displaystyle n+1}$

literature

• BW Gnedenko: Textbook of Probability Theory . Akademie Verlag, Berlin 1968, 1st German edition