Characteristic Function

The characteristic function of a subset of a larger set, sometimes also called the indicator function, is the function defined to be identically one on , and is zero elsewhere. Characteristic functions are sometimes denoted using the so-called Iverson bracket, and can be useful descriptive devices since it is easier to say, for example, "the characteristic function of the primes" rather than repeating a given definition. A characteristic function is a special case of a simple function.

The term characteristic function is used in a different way in probability. For any random variable , it is for real . When has a probability density function , this equals the Fourier transform of the probability density function using Fourier transform parameters . If for some , the following raw moment expansion is valid for :

			(1)
			(2)
			(3)
			(4)
			(5)

where (sometimes also denoted ) is the th raw moment and (Abramowitz and Stegun 1972, p. 928; Morrison 1995).

Every statistical distribution is uniquely specified by its characteristic function, even when its moments do not exist. A distribution need not be uniquely specified by all its moments. If is absolutely integrable, the distribution has a continuous probability density function given by

(6)

(Papoulis 1991, p. 116).

If , the characteristic function can be used to generate the th raw moment,

(7)

Under the exponential-moment hypothesis above, its logarithm, with the branch chosen so that , generates the cumulants near zero:

(8)