Characteristic Function

The characteristic function $\phi(t)$ is defined as the Fourier Transform of the Probability Density Function,

$$\phi(t) = {\mathcal F}[P(x)] = \int_{-\infty}^\infty e^{itx}P(x)\, dx$$ (1)

$$= \int_{-\infty}^\infty P(x)\,dx+it\int_{-\infty}^\infty xP(x)\,dx+{\textstyle{1\over 2}}(it)^2\int_{-\infty}^\infty x^2P(x)\,dx+\ldots$$ (2)

$$= \sum_{k=0}^\infty {(it)^k\over k!}\mu_k'$$ (3)

$$= 1+it\mu_1'-{\textstyle{1\over 2}}t^2\mu_2'-{\textstyle{1\over 3!}}it^3\mu'_3+{\textstyle{1\over 4!}}t^4\mu'_4+\ldots,$$ (4)

where $\mu_n'$ (sometimes also denoted $\nu_n$) is the $n$th Moment about 0 and $\mu_0'\equiv 1$. The characteristic function can therefore be used to generate Moments about 0,

$$\phi^{(n)}(0) \equiv \left[{d^n\phi\over dt^n}\right]_{t = 0} = i^n\mu'_n$$ (5)

or the Cumulants $\kappa_n$,

$$\ln\phi(t) \equiv \sum_{n=0}^\infty \kappa_n {(it)^n\over n!}.$$ (6)

A Distribution is not uniquely specified by its Moments, but is uniquely specified by its characteristic function.

See also Cumulant, Moment, Moment-Generating Function, Probability Density Function

References

Abramowitz, M. and Stegun, C. A. (Eds.). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, p. 928, 1972.

Kenney, J. F. and Keeping, E. S. ``Moment-Generating and Characteristic Functions,'' ``Some Examples of Moment-Generating Functions,'' and ``Uniqueness Theorem for Characteristic Functions.'' §4.6-4.8 in Mathematics of Statistics, Pt. 2, 2nd ed. Princeton, NJ: Van Nostrand, pp. 72-77, 1951.