Same here. Is the result mod 2pi? Unfortunately Wikipedia gives two definitions of the mod operator and the answer differs in the two cases. So you are right to be uncertain.nikozm said:I try to utilize this formula to a similar case, but the result seems too complicated. What if one of two RVs is a circular (mod 2 pi) uniformly distributed in [0, 2 pi) and the other one is an independent uniform RV in the range [-2^(-q) pi, 2^(-q) pi], where q is a nonnegative integer greater or equal than one. I presume that their sum is also a uniform RV, but I am not sure about its range.
Can you help me on this.
Thank you so much in advance.
Wikipedia : In computing, the modulo operation returns the remainder or signed remainder of a division, after one number is divided by another.
The sum of two independent circular uniform random variables, each uniformly distributed over the interval [0, 2π), is not uniformly distributed. Instead, the density function of their sum, \(Z = X + Y \mod 2\pi\), is characterized by a wrapped-around convolution of their individual uniform densities. The resulting distribution is known as a wrapped Cauchy distribution, which is more concentrated around certain angles depending on the sum.
The probability density function of the sum \(Z = X + Y \mod 2\pi\) of two independent circular uniform RVs can be calculated using convolution. The formula for the PDF is given by \( f_Z(z) = \frac{1}{2\pi} \int_0^{2\pi} f_X(x) f_Y(z-x \mod 2\pi) \, dx \), where \( f_X(x) = \frac{1}{2\pi} \) and \( f_Y(y) = \frac{1}{2\pi} \) are the densities of \(X\) and \(Y\). Simplifying, we get \( f_Z(z) = \frac{1}{2\pi} \), indicating the uniformity of distribution for each variable but not for their sum.
The characteristic function of a circular uniform random variable over [0, 2π) is given by \( \phi_X(t) = \frac{e^{it2\pi} - 1}{it2\pi} \). For the sum of two such independent variables, \(Z = X + Y\), the characteristic function is the product of the characteristic functions of \(X\) and \(Y\), hence \( \phi_Z(t) = \phi_X(t) \times \phi_Y(t) = \left(\frac{e^{it2\pi} - 1}{it2\pi}\right)^2 \). This function helps describe the distribution of \(Z\) and is useful in theoretical studies.
For circular statistics, the concepts of mean and variance are different from linear statistics. The mean direction of a circular uniform distribution is undefined as the distribution is symmetric around the circle. However, when considering the sum of two such variables, the resultant mean direction still remains undefined for similar reasons. The circular variance, which measures the concentration of data around the mean direction, is minimized in this case, indicating a higher concentration as compared to a single circular uniform variable.
As more circular uniform random variables are added, the sum tends to be more concentrated around certain angles. Mathematically, the density function of the sum of \(n\) independent circular uniform RVs increasingly resembles a von Mises distribution as \(n\) increases. This distribution is often referred to as the "circular normal" distribution because of its similarity to the Gaussian distribution in linear settings. The concentration parameter of the von Mises distribution increases with \(n\), indicating a sharper peak around the mean direction.