Arcsin distribution

Arcsin distribution
Density function
Distribution function
parameter	no
carrier	${\displaystyle x\in [0,1]}$
Density function	${\displaystyle {\frac {1}{\pi {\sqrt {x(1-x)}}}}}$
Distribution function	${\displaystyle {\frac {2}{\pi }}\arcsin \left({\sqrt {x}}\right)}$
Expected value	${\displaystyle {\frac {1}{2}}}$
Median	${\displaystyle {\frac {1}{2}}}$
mode	${\displaystyle \{0,1\}}$
Variance	${\displaystyle {\frac {1}{8}}}$
Crookedness	${\displaystyle 0}$

The arcsin distribution , also called the arcsine distribution , is a univariate probability distribution . It is a special case of the beta distribution with the parameters and plays an important role in the theory of Brownian motion . ${\displaystyle p=q={\tfrac {1}{2}}}$

definition

The Arcsin distribution is a probability distribution on . It is defined by its distribution function ${\displaystyle [0,1]}$

${\displaystyle F(x)={\frac {2}{\pi }}\arcsin \left({\sqrt {x}}\right),\ \ \ x\in [0,1]}$

and their probability density function

${\displaystyle f(x)={\begin{cases}{\frac {1}{\pi {\sqrt {x(1-x)}}}}&{\text{wenn}}\;x\in (0,1)\\0&{\text{wenn}}\;x\in \{0,1\}\\\end{cases}}}$.

properties

Let it be an arcsin-distributed random variable. ${\displaystyle X}$

Expectation and variance

The expected value results in

${\displaystyle \operatorname {E} (X)={\frac {1}{2}}}$

and the variance too

${\displaystyle \operatorname {Var} (X)={\frac {1}{8}}}$.

symmetry

The arcsin distribution is symmetrical around 0.5.

Arcsin Laws

There are a variety of Arcsin's laws. Publications on this come from, among others, Paul Lévy , Paul Erdős , Mark Kac and Erik Sparre Andersen . Some of the Arcsin laws are named after them.

The following Arcsin laws make statements about the duration of how long a stochastic process stays in the positive range. Instead, the images:

• earliest time of a maximum and
• the time when the origin was last crossed

must be considered, in which case further assumptions must then be made.

Arcsin Law by Paul Lévy

The lengths of time that a one-dimensional Standard Wiener process is positive are arcsin-distributed. That means for ${\displaystyle (W_{t})_{t\in [0,1]}}$

${\displaystyle U:=\lambda (\{t\in [0,1]\mid W_{t}>0\}),\ \ \ x\in [0,1]}$,

applies

${\displaystyle P(U\leq x)=F(x),\ \ \ x\in [0,1]}$,

where denotes the one-dimensional Lebesgue measure . ${\displaystyle \lambda }$

Arcsin law by Paul Erdős and Mark Kac

Let be a sequence of one-dimensional, independent and identically distributed random variables . It is also assumed that they have expectation value 0 and variance 1. The consecutive numbers of the sums ${\displaystyle (X_{i})_{i\in \mathbb {N} }}$

${\displaystyle S_{k}:=X_{1}+\dotsb +X_{k},\ \ \ k\in \mathbb {N} }$,

which are positive are defined by

${\displaystyle N_{n}:=|\{k\in \{1,\dotsc ,n\}\mid S_{k}>0\}|,\ \ \ n\in \mathbb {N} }$.

Then the following convergence in distribution holds

${\displaystyle \lim _{n\rightarrow \infty }P\left({\frac {N_{n}}{n}}\leq x\right)=F(x),\ \ \ x\in [0,1]}$.

The assumptions can be varied as long as the Central Limit Theorem continues to apply. ${\displaystyle (X_{i})_{i\in \mathbb {N} }}$

Arcsin law by Erik Sparre Andersen

Let be a sequence of random variables. The common densities exist for every selection of finitely many random variables and these are invariant with respect to s-permutations. An s-permutation consists of the composition of a permutation and a change of sign in any coordinates. Then, analogous to the Arcsin law of Erdős and Kac, the following convergence in distribution applies for the sums and the number of positive random variables ${\displaystyle (X_{i})_{i\in \mathbb {N} }}$ ${\displaystyle S_{k},\ k\in \mathbb {N} ,}$ ${\displaystyle N_{n},n\in \mathbb {N} ,}$

${\displaystyle \lim _{n\rightarrow \infty }P\left({\frac {N_{n}}{n}}\leq x\right)=F(x),\ \ \ x\in [0,1]}$.

Discrete arcsin distribution

Examples of the probability function of the discrete Arcsin distribution, where corresponds to a parameter and an expression. ${\displaystyle g_{n}(m)}$ ${\displaystyle n}$ ${\displaystyle m\in \{0,\dotsc ,n\}}$

In the fluctuation theory, Erik Sparre Andersen was able to show that the so-called discrete arcsin distribution is important. This is for each parameter by its probability function ${\displaystyle n\in \mathbb {N} \cup \{0\}}$

${\displaystyle g_{n}(m)=(-1)^{n}{\binom {-{\frac {1}{2}}}{m}}{\binom {-{\frac {1}{2}}}{n-m}},\ \ \ m\leq n}$

and their distribution function

${\displaystyle G_{n}(x)=\sum \limits _{m=0}^{\lfloor x\rfloor }(-1)^{n}{\binom {-{\frac {1}{2}}}{m}}{\binom {-{\frac {1}{2}}}{n-m}},\ \ \ x\in [0,n]}$

Are defined.

The name is justified by their convergence behavior to the Arcsin distribution, so the uniform convergence applies

${\displaystyle \lim _{n\rightarrow \infty }\sup _{x\in [0,1]}\mid G_{n}(n\cdot x)-F(x)\mid =0}$.

Erik Sparre Andersen showed the corresponding convergence in distribution in line with the previous Arcsin law.

literature

• William Feller: An introduction to probability theory and its applications . tape 2 . Wiley, 1971.
• Konrad Jacobs: Discrete Stochastics . Birkhäuser, Basel 2012, ISBN 3-0348-8645-4 .

Footnotes

1. ↑ Bauer, Heinz: Probability Theory . de Gruyter, 2002, p. 491-492 .
2. ^ Paul Lévy: Sur certains processus stochastiques homogènes, Compositio Mathematica . tape 7 , 1939, pp. 283-339 .
3. ^ Paul Erdős, Mark Kac: On the number of positive sums of independent random variables . In: Bull. Amer. Math. Soc. tape 53 , no. 10 , 1947, pp. 1011-1020 .
4. ↑ Erik Sparre Andersen: On the Number of Positive Sums of Random Variables . In: Scandinavian Actuarial Journal . tape 1949 , no. 1 , 1949, p. 27-36 , doi : 10.1080 / 03461238.1949.10419756 .