number: however, certain aspects of the theory of characteristic functions are advanced by extending the definition into the complex plane by analytical continuation, in cases where this is possible. Related concepts Related concepts include the moment-generating function and the probability-generating function. The characteristic function exists for all probability distributions. This is not the case for the moment-generating function. The characteristic function is closely related to the Fourier transform: the characteristic function of a probability density function p(x) is the complex conjugate of the continuous Fourier transform of p(x) (according to the usual convention; see continuous Fourier transform – other conventions). where P(t) denotes the continuous Fourier transform of the probability density function p(x). Likewise, p(x) may be recovered from φX(t) through the inverse Fourier transform: Indeed, even when the random variable does not have a density, the characteristic function may be seen as the Fourier transform of the measure corresponding to the random variable. Another related concept is the representation of probability distributions as elements of a reproducing kernel Hilbert space via the kernel embedding of distributions. This framework may be viewed as a generalization of the characteristic function under specific choices of the kernel function. See also Subindependence, a weaker condition than independence, that is defined in terms of characteristic functions. Cumulant, a term of the cumulant generating functions, which are logs of the characteristic functions. Notes References Citations Sources External links Functions related to probability distributions