Arcsin distribution
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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 .
definition
The Arcsin distribution is a probability distribution on . It is defined by its distribution function
and their probability density function
- .
properties
Let it be an arcsin-distributed random variable.
Expectation and variance
The expected value results in
and the variance too
- .
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
- ,
applies
- ,
where denotes the one-dimensional Lebesgue measure .
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
- ,
which are positive are defined by
- .
Then the following convergence in distribution holds
- .
The assumptions can be varied as long as the Central Limit Theorem continues to apply.
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
- .
Discrete arcsin distribution
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
and their distribution function
Are defined.
The name is justified by their convergence behavior to the Arcsin distribution, so the uniform convergence applies
- .
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
- ↑ Bauer, Heinz: Probability Theory . de Gruyter, 2002, p. 491-492 .
- ^ Paul Lévy: Sur certains processus stochastiques homogènes, Compositio Mathematica . tape 7 , 1939, pp. 283-339 .
- ^ 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 .
- ↑ 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 .