Continuously singular probability distribution

A continuously singular (probability) distribution is a special probability distribution in stochastics , which is characterized by its irregularity. Continuously singular probability distributions are neither represented by a probability function nor by a probability density function , but still have a continuous distribution function .

Continuously singular distributions rarely occur or have to be specially constructed. An example of this is the Cantor distribution .

definition

A probability distribution is given . ${\displaystyle P}$ ${\displaystyle (\mathbb {R} ,{\mathcal {B}}(\mathbb {R} ))}$

Then a continuously singular probability distribution is called if is an atomless measure and is singular with respect to the Lebesgue measure . ${\displaystyle P}$ ${\displaystyle P}$

This means in full:

• For everyone is (atomless) ${\displaystyle x\in \mathbb {R} }$ ${\displaystyle P(\{x\})=0}$
• There is an with and (singularity) ${\displaystyle A\in {\mathcal {B}}(\mathbb {R} )}$ ${\displaystyle P(\mathbb {R} \setminus A)=0}$ ${\displaystyle \lambda (A)=0}$

example

Plot of the Cantor function (10 iterations)

A typical example of a continuously singular distribution is the Cantor distribution , the distribution function of which is shown on the right. The exact construction is explained in the main article on Cantor distribution and is closely related to the Cantor set . ${\displaystyle {\mathcal {C}}}$

It should be noted that the distribution function is continuous, from which it follows that the Cantor distribution has no discrete part or is atomless. Because every atom, i.e. every atom with , would express itself as a jump point in the distribution function. ${\displaystyle x}$ ${\displaystyle P(\{x\})>0}$

Furthermore, due to its construction, the distribution function is constant on the complement of the Cantor set . It follows that . However, since the Cantor set itself has the Lebesgue measure 0, that is, the Cantor distribution and the Lebesgue measure are singular to one another. ${\displaystyle {\mathcal {C}}}$ ${\displaystyle P(\mathbb {R} \setminus {\mathcal {C}})=0}$ ${\displaystyle \lambda ({\mathcal {C}})=0}$

Thus, the Cantor distribution is atomless and singular to Lebesgue measure, i.e. continuously singular.

properties

• As already mentioned above, a continuously singular distribution has neither a probability density function nor a probability function, but a continuous distribution function.
• Due to the non-existence of the probability (density) function, the mode does not exist.
• According to the Lebesgue decomposition , each probability distribution can be broken down into a discrete probability distribution (with probability function), an absolutely continuous probability distribution (with probability density function) and a continuously singular probability distribution.

Web links

• VG Ushakov: Singular distribution . In: Michiel Hazewinkel (Ed.): Encyclopaedia of Mathematics . Springer-Verlag , Berlin 2002, ISBN 978-1-55608-010-4 (English, online ).

literature

• Jürgen Elstrodt : Measure and integration theory . 6th, corrected edition. Springer-Verlag, Berlin Heidelberg 2009, ISBN 978-3-540-89727-9 , doi : 10.1007 / 978-3-540-89728-6 .
• Klaus D. Schmidt: Measure and Probability . 2nd, revised edition. Springer-Verlag, Heidelberg Dordrecht London New York 2011, ISBN 978-3-642-21025-9 , doi : 10.1007 / 978-3-642-21026-6 .

Individual evidence

1. ^ Schmidt: Measure and probability. 2011, p. 259.