Singular Distribution: Definition and Examples

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Singular distribution definition

A singular distribution is concentrated on a set of Lebesgue measure zero. This means that the probability of any point in the set is zero. A singularity is a point where a function or a measure is undefined; in the context of probability, a singularity can be thought of as a point where the distribution is undefined. This can happen for a variety of reasons, such as when the distribution is concentrated on a set of points that have zero measure.

An alternate definition of a singular distribution is one that is singular with respect to the Lebesgue measure and has no atoms [1]. An atom is a set of points that cannot be broken down into smaller sets with probability greater than zero.

A singular distribution can also be described as one that has no absolutely continuous or discrete part [2]. It doesn’t have a probability density function, since the Lebesgue integral of any such function would be zero; it is not a discrete probability distribution (pdf) because each discrete point has a zero probability. Thus, a singular distribution can’t be represented as a sum of a continuous pdf and a discrete probability mass function (PMF). On the other hand, a regular distribution can be represented as a sum of a continuous PDF and a discrete PMF. Regular distributions comprise most of the probability distributions you’ll come across in statistics.

Singular distributions are sometimes called singular continuous distributions, because their cumulative distribution functions (CDFs) are singular and continuous.

Singular distribution examples

 

Singular distributions aren’t that common in probability, but there are a few notable ones:

• The Cantor Distribution is a probability distribution whose CDF is the Cantor Function. The Cantor function is an example of a pathological function that is horizontal almost everywhere yet always climbs upwards. This means that the probability of any point in the Cantor set is zero.
• A continuous degenerate distribution is a special case of a singular distribution [4]. It is a singular distribution that is also continuous.
• Dirac delta distribution: The Dirac delta is an element of a set of mathematical objects called distributions – so the function is more aptly named a “delta distribution.” It is a singular distribution because its pdf is not integrable [5].

The pdf of a singular distribution can be undefined at some points, but that doesn’t mean a singular distribution has an undefined pdf. For example, the Cauchy distribution is a singular distribution, but its pdf is defined at all points.

References:

[1] Žitkovic, G. Theory of Probability: Measure theory, classical probability and stochastic analysis. Lecture Notes

[2] Wolpert, R. STA 711: Probability & Measure Theory

[3] Image: CantorEscalier.svg: Theonderivative work: Amirki, CC BY-SA 3.0 https://creativecommons.org/licenses/by-sa/3.0, via Wikimedia Commons

[4] Gentle, J. (2003). Theory of Statistics.

[5] Howard, R. (2012). Dirac Delta and Singular Distributions: The General Non-good Function Case. Conference: International Conference on Engineering and Applied Science At: Beijing.