On the real line , the definition of a singular distribution is equivalent to the following: A distribution is singular if the corresponding distribution function is continuous and its set of growth points has Lebesgue measure zero.
An example of a singular distribution on is a distribution concentrated on the Cantor set, the so-called Cantor distribution, which can be described in the following way. Let be a sequence of independent random variables, each of which takes on the values 0 and 1 with probability . Then the random variable
has a Cantor distribution, and its characteristic function is equal to
An example of a singular distribution on () is a uniform distribution on a sphere of positive radius.
The convolution of two singular distributions can be singular, absolutely continuous or a mixture of the two.
Any probability distribution can be uniquely represented in the form
where is discrete, is absolutely continuous, is singular, , and (Lebesgue decomposition).
Sometimes, singularity is understood in a wider sense: A probability distribution is singular with respect to a measure if it is concentrated on a set with . Under this definition, every discrete distribution is singular with respect to Lebesgue measure.
For singular set functions, see also Absolute continuity of set functions.
|||Yu.V. [Yu.V. Prokhorov] Prohorov, Yu.A. Rozanov, "Probability theory, basic concepts. Limit theorems, random processes", Springer (1969) (Translated from Russian)|
|||W. Feller, "An introduction to probability theory and its applications", 2, Wiley (1971)|
Singular distribution. Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Singular_distribution&oldid=25977