is to the theory of the decomposability of random variables. Definition For a scalar random variable X the characteristic function is defined as the expected value of eitX, where i is the imaginary unit, and is the argument of the characteristic function: Here FX is the cumulative distribution function of X, and the integral is of the Riemann–Stieltjes kind. If a random variable X has a probability density function fX, then the characteristic function is its Fourier transform with sign reversal in the complex exponential. QX(p) is the inverse cumulative distribution function of X also called the quantile function of X. This convention for the constants appearing in the definition of the characteristic function differs from the usual convention for the Fourier transform. For example, some authors define , which is essentially a change of parameter. Other notation may be encountered in the literature: as the characteristic function for a probability measure p, or as the characteristic function corresponding to a density f. Generalizations The notion of characteristic functions generalizes to multivariate random variables and more complicated random elements. The argument of the characteristic function will always belong to the continuous dual of the space where the random variable X takes its values. For common cases such definitions are listed below: If X is a k-dimensional random vector, then for where is the transpose of the vector , If X is a k × p-dimensional random matrix, then for where is the trace operator, If X is a complex random variable, then for where is the complex conjugate of and is the real part of the complex number , If X is a k-dimensional complex random vector, then for where is the conjugate transpose of the vector , If X(s) is a stochastic process, then for all functions t(s) such that the integral converges for almost all realizations of X Examples Oberhettinger (1973) provides extensive tables of characteristic functions. Properties The characteristic function of a real-valued random variable always exists, since it is an integral of a bounded continuous function over a space whose measure is finite. A characteristic function is uniformly continuous on the entire space. It is non-vanishing in a region around zero: φ(0) = 1. It is bounded: |φ(t)| ≤ 1. It is Hermitian: . In particular, the characteristic function of a symmetric (around the origin) random variable is real-valued and even. There is a bijection between