Dirac Comb - Use in Directional Statistics

Use in Directional Statistics

In directional statistics, the Dirac comb of period 2π is equivalent to a wrapped Dirac delta function, and is the analog of the Dirac delta function in linear statistics.

In linear statistics, the random variable (x) is usually distributed over the real number line, or some subset thereof, and the probability density of x is a function whose domain is the set real numbers, and whose integral from to is unity. In directional statistics, the random variable (θ) is distributed over the unit circle and the probability density of θ is a function whose domain is some interval of the real numbers of length 2π and whose integral over that interval is unity. Just as the integral of the product of a Dirac delta function with an arbitrary function over the real number line yields the value of that function at zero, so the integral of the product of a Dirac comb of period 2π with an arbitrary function of period 2π over the unit circle yields the value of that function at zero.