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characteristic function[‚kar·ik·tə′ris·tik ′fəŋk·shən]
(1) An eigenfunction.
(2) The characteristic, or indicator, function of a set A is a function f(x) that is defined on some set E containing A and that assumes the value f(x) = 1 if x is in A and the value f(x) = 0 if x is not in A.
(3) In probability theory, the characteristic function fx(t) of a random variable X is the mathematical expectation of the quantity exp (it X). For a random variable with probability density function PX(x), this definition yields the formula
For example, for a random variable having a normal distribution with parameters a and σ, the characteristic function is
The characteristic function has several noteworthy properties. To every random variable X there corresponds a definite characteristic function. The probability distribution for X is uniquely determined by fX(t). When independent random variables are added, the corresponding characteristic functions are multiplied. If the concept of closeness is suitably defined, to random variables with close distributions there correspond characteristic functions that differ little from each other, and, conversely, to close characteristic functions there correspond random variables with close distributions. These properties underlie the applications of characteristic functions, particularly applications to the derivation of the limit theorems of probability theory.
A mathematical apparatus more or less equivalent to that of the characteristic function was first used by P. Laplace in 1812, but the full power of the characteristic function method was demonstrated in 1911 by A. M. Liapunov, who used the method in obtaining the theorem that bears his name.
The concept of characteristic function can be generalized to finite and infinite systems of random variables—that is, to random vectors and stochastic processes. The theory of characteristic functions has much in common with the theory of Fourier integrals.
REFERENCESGnedenko, B. V. Kursteorii veroiatnostei, 5th ed. Moscow, 1969.
Prokhorov, Iu. V., and Iu. A. Rozanov. Teoriia veroiatnostei, 2nd ed. Moscow, 1973.