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A [[Probability distribution|probability distribution]] on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085530/s0855301.png" /> concentrated on a set of [[Lebesgue measure|Lebesgue measure]] zero and giving probability zero to every one-point set.
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A [[Probability distribution|probability distribution]] on $\mathbf R^n$ concentrated on a set of [[Lebesgue measure|Lebesgue measure]] zero and giving probability zero to every one-point set.
  
On the real line <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085530/s0855302.png" />, 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.
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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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085530/s0855303.png" /> is a distribution concentrated on the Cantor set, the so-called Cantor distribution, which can be described in the following way. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085530/s0855304.png" /> be a sequence of independent random variables, each of which takes on the values 0 and 1 with probability <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085530/s0855305.png" />. Then the random variable
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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
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085530/s0855306.png" /></td> </tr></table>
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$$Y=2\sum_{j=1}^\infty\frac{1}{3^j}X_j$$
  
 
has a Cantor distribution, and its characteristic function is equal to
 
has a Cantor distribution, and its characteristic function is equal to
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085530/s0855307.png" /></td> </tr></table>
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$$f(t)=e^{it/2}\prod_{j=1}^\infty\cos\frac{t}{3^j}.$$
  
An example of a singular distribution on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085530/s0855308.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085530/s0855309.png" />) is a [[Uniform distribution|uniform distribution]] on a sphere of positive radius.
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An example of a singular distribution on $\mathbf R^n$ ($n\geq2$) is a [[Uniform distribution|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.
 
The convolution of two singular distributions can be singular, absolutely continuous or a mixture of the two.
  
Any probability distribution <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085530/s08553010.png" /> can be uniquely represented in the form
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Any probability distribution $P$ can be uniquely represented in the form
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085530/s08553011.png" /></td> </tr></table>
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$$P=a_1P_d+a_2P_a+a_3P_s,$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085530/s08553012.png" /> is discrete, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085530/s08553013.png" /> is absolutely continuous, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085530/s08553014.png" /> is singular, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085530/s08553015.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085530/s08553016.png" /> (Lebesgue decomposition).
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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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085530/s08553017.png" /> is singular with respect to a measure <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085530/s08553018.png" /> if it is concentrated on a set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085530/s08553019.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085530/s08553020.png" />. Under this definition, every discrete distribution is singular with respect to Lebesgue measure.
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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|Absolute continuity]] of set functions.
 
For singular set functions, see also [[Absolute continuity|Absolute continuity]] of set functions.
  
 
====References====
 
====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> Yu.V. [Yu.V. Prokhorov] Prohorov,   Yu.A. Rozanov,   "Probability theory, basic concepts. Limit theorems, random processes" , Springer (1969) (Translated from Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> W. Feller,   "An introduction to probability theory and its applications" , '''2''' , Wiley (1971)</TD></TR></table>
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<table><TR><TD valign="top">[1]</TD> <TD valign="top"> Yu.V. [Yu.V. Prokhorov] Prohorov, Yu.A. Rozanov, "Probability theory, basic concepts. Limit theorems, random processes", Springer (1969) (Translated from Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> W. Feller, [[Feller, "An introduction to probability theory and its applications"|"An introduction to probability theory and its  applications"]], '''2''', Wiley (1971)</TD></TR></table>

Latest revision as of 18:13, 3 August 2014

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)
How to Cite This Entry:
Singular distribution. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Singular_distribution&oldid=19272
This article was adapted from an original article by V.G. Ushakov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article