Wrapped Normal Distribution - Definition

Definition

The probability density function of the wrapped normal distribution is

f_{WN}(\theta;\mu,\sigma)=\frac{1}{\sigma \sqrt{2\pi}} \sum^{\infty}_{k=-\infty} \exp \left

where μ and σ are the mean and standard deviation of the unwrapped distribution, respectively. Expressing the above density function in terms of the characteristic function of the normal distribution yields:

f_{WN}(\theta;\mu,\sigma)=\frac{1}{2\pi}\sum_{n=-\infty}^\infty e^{-\sigma^2n^2/2+in(\theta-\mu)} =\frac{1}{2\pi}\vartheta\left(\frac{\theta-\mu}{2\pi},\frac{i\sigma^2}{2\pi}\right) ,

where is the Jacobi theta function, given by

\vartheta(\theta,\tau)=\sum_{n=-\infty}^\infty (w^2)^n q^{n^2} \text{ where } w \equiv e^{i\pi \theta} and

The wrapped normal distribution may also be expressed in terms of the Jacobi triple product:

where and

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