Cantor distribution

The Cantor distribution is a probability distribution that is characterized by the fact that it has neither a probability density function nor a probability function , but is continuously singular . The associated distribution function is known as the Cantor function or the devil's staircase .

Plot of the Cantor function (10 iterations)

construction

The Cantor distribution (also known as Borel's σ-algebra ) cannot simply be given explicitly. It has to be constructed recursively, similar to the Cantor set . ${\displaystyle \mu :{\mathcal {B}}(\mathbb {R} )\rightarrow [0,1]}$ ${\displaystyle {\mathcal {B}}(\mathbb {R} )}$

1st variant

If one starts from the uniformly distributed measure on the quantity , one obtains a product measure on the quantity . This measure can be interpreted as follows: Consider an experiment in which a fair coin is tossed infinitely often; Elements of can be interpreted as the outcome of the experiment (the sequence means, for example, that heads and tails always appeared alternately). From now on, the measure assigns its probability to a subset . For example, the strong law of large numbers says that the set of "uniformly distributed" sequences has probability 1, where the following set is: ${\displaystyle \{0,1\}}$ ${\displaystyle 2^{\mathbb {N} }}$ ${\displaystyle \mu }$ ${\displaystyle 2^{\mathbb {N} }}$ ${\displaystyle (0,1,0,1,\ldots )}$ ${\displaystyle \mu }$ ${\displaystyle 2^{\mathbb {N} }}$ ${\displaystyle G}$ ${\displaystyle G}$

${\displaystyle G=\left\{(x_{0},x_{1},\ldots )\left|\ \lim \limits _{n\to \infty }{\frac {|\{i<n:x_{i}=0\}|}{n}}={\frac {1}{2}}\right.\right\}.}$

Now the Cantor set - as explained in the article there - can be mapped bijectively on . By means of this bijection , the measure mentioned above can be converted into a probability measure on the Cantor set. (An alternative description of results from the Hausdorff measure to the dimension .) ${\displaystyle C}$ ${\displaystyle 2^{\mathbb {N} }}$ ${\displaystyle \mu }$ ${\displaystyle \mu }$ ${\displaystyle \mu }$ ${\displaystyle \ln 2/\ln 3}$

This probability measure is the Cantor distribution, an example of a measure whose distribution function is continuous, but not absolutely continuous . The distribution function ${\displaystyle \mu }$

${\displaystyle {\begin{aligned}F:[0,1]&\to [0,1]\\x&\mapsto \mu ([0,x]\cap C)\end{aligned}}}$

is called Cantor function (also "Cantor's staircase function"). This function is constant on every interval in the complement of the Cantor set; on the interval has, for example, the value of 1/2, and on the interval it has the value 1/4. ${\displaystyle \left({\tfrac {1}{3}},{\tfrac {2}{3}}\right)}$ ${\displaystyle \left({\tfrac {1}{9}},{\tfrac {2}{9}}\right)}$

2nd variant

With this construction the Cantor function is constructed, which uniquely determines the Cantor distribution according to the correspondence sentence . ${\displaystyle F:\mathbb {R} \rightarrow [0,1]}$ ${\displaystyle \mu }$

Let be the system of all subsets of , which can be represented as the union of finitely many disjoint closed non-empty intervals. Further be given by (with ) ${\displaystyle {\mathcal {G}}}$ ${\displaystyle [0,1]}$ ${\displaystyle \varphi :{\mathcal {G}}\rightarrow {\mathcal {G}}}$ ${\displaystyle i,m\in \mathbb {N} ,a_{i},b_{i}\in [0,1],a_{i}\leq b_{i}}$

${\displaystyle \varphi \left(\bigcup _{i\in \{1,\ldots ,m\}}[a_{i},b_{i}]\right):=\bigcup _{i\in \{1,\ldots ,m\}}\left(\left[a_{i},{\frac {2a_{i}+b_{i}}{3}}\right]\cup \left[{\frac {a_{i}+2b_{i}}{3}},b_{i}\right]\right)}$

(This corresponds to the already mentioned recursive division into thirds of the intervals (interval length:) , whereby only the lower and the upper third are taken while the middle third is "wiped out".) ${\displaystyle b_{i}-a_{i}}$

Keep on being with ${\displaystyle n\in \mathbb {N} }$

${\displaystyle C_{n}:=\varphi ^{n}([0,1])\,.}$

Finally, let the Cantor set be defined by ${\displaystyle C}$

${\displaystyle C=\bigcap _{n\in \mathbb {N} }C_{n}\,.}$

Now the dimension is defined as follows: ${\displaystyle \mu _{n}:{\mathcal {B}}(\mathbb {R} )\rightarrow [0,1]}$

${\displaystyle \mu _{n}:=\int \left({\frac {3}{2}}\right)^{n}\chi _{C_{n}}(x)d\lambda (x)}$,

where denotes the one-dimensional Lebesgue measure. is obviously a probability measure which is the associated distribution function . The following applies to: ${\displaystyle \lambda }$ ${\displaystyle \mu _{n}}$ ${\displaystyle F_{n}:\mathbb {R} \rightarrow [0,1]}$ ${\displaystyle F_{n}}$

${\displaystyle F_{n}(x)=\left({\frac {3}{2}}\right)^{n}\lambda (C_{n}\cap [0,x])}$

For especially and . ${\displaystyle F_{n}}$ ${\displaystyle F_{n}(0)=0}$ ${\displaystyle F_{n}(1)=1}$

Since is uniformly convergent , the Cantor function is through ${\displaystyle F_{n}|_{[0,1]}}$ ${\displaystyle F}$

${\displaystyle F(x):={\begin{cases}0&{\text{falls }}x\in (-\infty ,0]\\\lim \limits _{n\rightarrow \infty }F_{n}|_{[0,1]}(x)&{\text{falls }}x\in (0,1)\\1&{\text{falls }}x\in [1,\infty )\end{cases}}}$

clearly defined. The associated distribution in the sense of measure theory is the Cantor distribution.

properties

• The Cantor distribution is singular with respect to the Lebesgue measure.
• The Cantor distribution is a symmetrical distribution .
• The Cantor distribution has no Lebesgue density .
• The Cantor function is steadily and monotonically increasing between 0 and 1.
• The Cantor function is almost everywhere differentiable with a derivative of 0, but is still not constant.

In integration theory, expressions of form result

${\displaystyle \int _{0}^{1}g(x)\,{\rm {d}}F(x),}$

where a bounded measurable function on the interval is a sense, not an expression of the form ${\displaystyle g}$ ${\displaystyle [0,1]}$

${\displaystyle \int _{0}^{1}g(x)\,{\frac {{\rm {d}}F}{{\rm {d}}x}}(x)\,\mathrm {d} x.}$

Physical realizations

In physics, devil's stairs occur approximately in systems with competing lengths (e.g. in adsorbates or with structural phase transitions , which are described by the model of Frenkel and Kontorowa) or with competing interactions (e.g. magnets or alloys , which are caused by the ANNNI model ). Devil's Stairs could also describe the temporally "clumped" occurrence of earthquakes.

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 .

Individual evidence

1. ↑ Nadja Podbregar: Earthquakes follow “devil's stairs” . In: scinexx | The knowledge magazine . April 15, 2020 ( scinexx.de [accessed April 15, 2020]).