Distributions on the circle group

Abstract. In this paper, we extend the definition of a random angle and the definition of a probability distribution of a random angle. We expand P. Lévy’s researches related to wrapping the probability distributions defined on R. We determine a relation between quasi-lattice probability distributions on R and lattice probability distributions on the unit circle S. We use the Bergström identity for comparison of a convolution of probability distributions of random angles. We also prove an inverse formula for lattice probability distributions on S.


Preface
Von Mises [17], Perrin [15] and Fisher [6] have made the start of the new trend in statistics.In present, it is called a statistical analysis of directional observations.In 1939, Lévy [11] published methods useful analyzing distributions on torus T using the results of probability theory distributions on the real line R.
In the encyclopedic publication "Probability measures on locally compact groups", in 1977, Heyer has noticed that the theory of Gaussian distribution on the unit circle was developed by von Mises [17], Lévy [11] and it was widely discussed in Mardia's monograph [13], which was translated into Russian in 1978 [12].Mardia's book is useful to solve statistical problems when probability distributions are on the unit circle.The developed theory is of great significance in various applications, for example, in spectroscopy, geodesy, navigation etc. Mardia book is a comprehensive monograph useful to practitioners and for all whose researches are related to probability distributions on the unit circle.Later, in 1999, Mardia and Jupp published a monograph [14] in which one can find more well-founded statements about directional statistics.In 2013, Pewsey, Neuhauser, Ruxton [16] published a book useful in working with software environment R.
In the translation of Mardia book [12], editor L.N. Bolshev submitted notes about definitions of random variables and their distributions.In this paper, definitions that did not cause discussion questions are used.We also consider the subjects related to probability distributions on locally compact Abelian (LCA) groups [4,7,18].
In the articles of statisticians and probability theory specialists, distributions on the unit circle S = cos θ, sin θ : 0 θ < 2π , the centre of which is at the origin, are constructed not only directly on the unit circle, but also using the following methods: wrapping, offsetting, characterizing and stereographic projection [1,6,9].The methods are based on probability distributions on the real line R or on the space R 2 .In this paper, we do not discuss the offsetting method.The main results are related to the wrapped probability distributions and probability distributions constructed directly on S.

Preliminaries
Suppose, {Ω, F} is a measurable space.Definition 1.We say Θ = Θ(ω) is a random angle given on measurable space {Ω, F} if for every Borel set B ∈ B(S), Thus, the random angle Θ generates measurable space {S, B(S)}, where S is the unit circle, and B(S) is σ-algebra of Borel sets generated by S. Suppose, P denotes a probability on {Ω, F}.
which satisfies the equality One can find the definition of a probability function of a random angle and its properties in [13,14] For −∞ < α < β < ∞ and β − α < 2π, where the integral is a Lebesgue-Stieltjes integral. https://www.mii.vu.lt/NA The main results of the paper are related to distributions of a lattice random angle.Before introducing its definition, we remind the definition of lattice and quasi-lattice random variables.
where a ∈ R and h > 0.
Esseen [5] was the first who widely used lattice random variables in the theory of sums of independent random variables.He proves an inverse formula where In [2], a quasi-lattice random variable is defined.One can also find an inverse formula of the same type as (1) for the quasi-lattice random variable in [2].

Definition 5. A random variable η is quasi-lattice if it takes its values in
where β 1 , β 2 > 0 are rationally independent, i.e.
In [2], the inverse formula obtained, where (γ 1 , γ 2 ) is a two-dimensional lattice random vector, and its characteristic function is The random vector (γ 1 , γ 2 ) is related to the random variable η: One can find the definition of distributions of lattice random angles in Mardia book [13] on page 54.Definition 6.The distribution of a random angle Θ is called lattice if for some l 1, We write P{Θ = 2πr/l} instead of P{Θ = 2πr/l (mod 2π)}.
It follows from the definition above that a characteristic function of a lattice random angle Θ is For instance, Mardia [13, p. 54] calls the random angle Θ as Poisson if it takes the values where l is integer and l 1, with probabilities Let ξ be a random variable defined on the real line R, and let F ξ (x), x ∈ R, be its probability function.
Now we present some properties of wrapping that can be found in [13,14].
(i) Wrapping is a homomorphism from R to S: operation + in the right-hand side of the equality denotes addition modulo 2π.(ii) If ξ is a random variable and Θ w = ξ (mod 2π) is a random angle, then their characteristic functions are related by Ee ipΘw = Ee itξ t=p , p = 0, ±1, ±2, . . . .
(iii) If a random variable ξ is infinitely divisible, then a random angle Θ w = ξ (mod 2π) is also infinitely divisible.
One can also wrap the probabilities of a random variable, which takes integer values.If ξ is a random variable, which takes values m = 0, ±1, ±2, . . .with probabilities P{ξ = m}, then the random angle Θ w = 2πξ/l (mod 2π) takes its values in the lattice 2π l r: r = 0, 1, . . ., l − 1 , l 1, with probabilities Lévy [11] obtained the definition of Poisson distribution on S by wrapping the Poisson distribution on the circumference of the unit circle S with the centre at the origin.One can find a wrapped t-distribution in [10], a wrapped classic exponential distribution and the Laplace probability distribution in [8].Thus, the wrapping relates the probability distributions on S to probability distributions on R.
The main results of this paper are related to the wrapped lattice and quasi-lattice probability distributions.

Main results
Suppose, we have a random variable ξ and a random angle Θ w = ξ (mod 2π) given on probability space {Ω, F, P}.First of all, we are going to make an important notice, which enables us to write the wrapped distribution function in a useful form.So, we can use the set in equality ( 4) and to write the wrapped distribution function as follows: Remark 1.In equality ( 5), the defined Borel set B(θ) is a union of disjoint intervals Thus, it follows from equality (6) that A natural question arises whether we get the probability distribution of a lattice random angle by wrapping the probability distribution of a quasi-lattice random variable.A hypothesis would be that we obtain the Haar probability distribution.
Theorem 1. Assume that ξ is a random variable taking values in where (β 1 , β 2 ) is the integer basis.After wrapping the probability function of the random variable ξ on the unit circle S, we obtain the probability distribution of random angle Θ ξ , which takes its values with probabilities for all r = 0, 1, . . ., l − 1 and any integer l 1. https://www.mii.vu.lt/NA We use formula (7) to obtain the Bergström identity on the unit circle S.
Proof of Theorem 1.It is obvious that Let us refer to the characteristic function of a quasi-lattice random variable on R, i.e. the function We can rewrite it in another form using equality (12).Let Then Nonlinear Anal.Model.Control, 24(3):433-446 Let us substitute p = 0, ±1, ±2, . . .for t in the previous equality.Therefore Note that β 1 and β 2 are rationally independent, i.e.
Let us substitute t ∈ R for p in equality (8).We obtain In the previous equality, the number series absolutely converges for all t ∈ R. Consequently, we take t = lt , t ∈ R, in equality (9).Then we substitute p = 0, ±1, ±2, . . .for t .Hence Once again we substitute t ∈ R for p in equality (10) and obtain In equality (11), it is useful to choose and then to substitute p = 0, ±1, ±2, . . .for t .We get On the right-hand side of the previous equality, there is the characteristic function of random angle Θ ξ = ξ (mod 2π), and Now we must verify the conditions of the definition of a lattice random angle on S.
This equality follows from the fact that ξ is a quasi-lattice random variable.
The proof of Theorem 1 is complete.
Remark 3. In the proof of Theorem 1, one can find a method how to wrap the probability distribution on S.
Corollary 1. Suppose that a random variable ξ takes values hν, ν = 0, ±1, ±2, . . ., h > 0, and Θ w = 2πξ/lh (mod 2π), l 1 is integer, h > 0. For any h > 0 and any integer l 1, the following equality is true: Nonlinear Anal.Model.Control, 24(3):433-446 The statement of Corollary 1 is easy to see, but in proving it, the point is to demonstrate a method how to wrap the probability distribution of a random variable on the circumference of the unit circle.
Proof of Corollary 1. From the definition of the random variable ξ it follows that It is easy to see that where l is the integer and l 1.Thus, the characteristic function of the random variable ξ is In formula (13), let us choose for all r = 0, 1, . . ., l − 1 and any integer l 1.
The proof of Corollary 1 is complete.
Corollary 2. The values of lattice random angle Θ w = ξ (mod 2π) do not depend on the span h of the distribution of lattice random variable ξ.
Let us take probability distributions of two quasi-lattice random angles ξ and η defined in the integer base where β 1 , β 2 are rationally independent.https://www.mii.vu.lt/NA

Definition 2 .
We call the function P Θ (B) = P ω: Θ(ω) ∈ B of all Borel sets B ∈ B(S) a probability distribution of random angle Θ given on the space {S, B(S)}.Definition 3. The function