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Does there exist a probability density distribution function with analogous properties to those of the Gaussian distribution, but defined on domains such as:

  1. a limited segment $$x \in [a,b]$$
  2. a circular domain (a domain which "wraps around") $$\theta \in [0, 2\pi)$$

By asking "properties similar to the Gaussian", I would ideally like something as the central limit theorem.

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For 2. check out von Mises distribution also known as the circular normal distribution. –  Sasha Dec 21 '12 at 13:17
    
You might also want to look at central limit theorems in free probability. –  Eckhard Dec 21 '12 at 13:21
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