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In probability theory, the characteristic function of any random variable completely defines its probability distribution. On the real line it is given by the following formula, where X is any random variable with the distribution in question:
In cases in which there is a probability density function, fX, this becomes
Every probability distribution on R or on Rn has a characteristic function, because one is integrating a bounded function over a space whose measure is finite.
The inversion theoremEdit
More than that, there is a bijection between cumulative probability distribution functions and characteristic functions. In other words, two distinct probability distributions never share the same characteristic function.
Given a characteristic function φ, it is possible to reconstruct the corresponding cumulative probability distribution function F:
The continuity theoremEdit
If the sequence of characteristic functions of distributions Fn converges to the characteristic function of a distribution F, then Fn(x) converges to F(x) at every value of x at which F is continuous.
Uses of characteristic functionsEdit
Characteristic functions are particularly useful for dealing with functions of independent random variables. For example, if X1, X2, ..., Xn is a sequence of independent (and not necessarily identically distributed) random variables, and
where the ai are constants, then the characteristic function for Sn is given by
In particular, . To see this, write out the definition of characteristic function:
Observe that the independence of and is required to establish the equality of the third and fourth expressions.
Because of the continuity theorem, characteristic functions are used in the most frequently seen proof of the central limit theorem.
Characteristic functions can also be used to find moments of random variable. Provided that nth moment exists, characteristic function can be differentiated n times and
Characteristic functions arise in the statement and proof of Bochner's theorem.
Related concepts include the moment-generating function and the probability-generating function. The characteristic function exists for all probability distributions. However this is not the case for moment generating function.
The characteristic function is closely related to the Fourier transform: the characteristic function of a probability density function is the complex conjugate of the continuous Fourier transform of (according to the usual convention; see ).
where denotes the continuous Fourier transform of the probability density function . Likewise, may be recovered from 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.
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