Singular distribution

A probability distribution on $\mathbf R^n$ concentrated on a set of Lebesgue measure zero and giving probability zero to every one-point set.

On the real line $\mathbf R^1$, 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 $\mathbf R^1$ is a distribution concentrated on the Cantor set, the so-called Cantor distribution, which can be described in the following way. Let $X_1,X_2,\ldots,$ be a sequence of independent random variables, each of which takes on the values 0 and 1 with probability $1/2$. Then the random variable

$$Y=2\sum_{j=1}^\infty\frac{1}{3^j}X_j$$

has a Cantor distribution, and its characteristic function is equal to

$$f(t)=e^{it/2}\prod_{j=1}^\infty\cos\frac{t}{3^j}.$$

An example of a singular distribution on $\mathbf R^n$ ($n\geq2$) 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 $P$ can be uniquely represented in the form

$$P=a_1P_d+a_2P_a+a_3P_s,$$

where $P_d$ is discrete, $P_a$ is absolutely continuous, $P_s$ is singular, $a_i\geq0$, and $a_1+a_2+a_3=1$ (Lebesgue decomposition).

Sometimes, singularity is understood in a wider sense: A probability distribution $F$ is singular with respect to a measure $P$ if it is concentrated on a set $N$ with $P\{N\}=0$. Under this definition, every discrete distribution is singular with respect to Lebesgue measure.

For singular set functions, see also Absolute continuity of set functions.

References

[1]	Yu.V. [Yu.V. Prokhorov] Prohorov, Yu.A. Rozanov, "Probability theory, basic concepts. Limit theorems, random processes", Springer (1969) (Translated from Russian)
[2]	W. Feller, "An introduction to probability theory and its applications", 2, Wiley (1971)