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Characteristic Function

Characteristic Function#

The library makes heavy use of characteristic function concept and therefore, it is useful to familiarize with it.

Definition#

The characteristic function of a random variable \(x\) is the Fourier (inverse) transform of \(P^x\), where \(P^x\) is the distrubution measure of \(x\)

(10)#\[\begin{equation} \Phi_{x,u} = {\mathbb E}\left[e^{i u x_t}\right] = \int e^{i u x} P^x\left(dx\right) \end{equation}\]

Properties#

  • \(\Phi_{x, 0} = 1\)

  • it is bounded, \(\left|\Phi_{x, u}\right| \le 1\)

  • it is Hermitian, \(\Phi_{x, -u} = \overline{\Phi_{x, u}}\)

  • it is continuous

  • characteristic function of a symmetric random variable is real-valued and even

  • moments of \(x\) are given by

(11)#\[\begin{equation} {\mathbb E}\left[x^n\right] = i^{-n} \left.\frac{\Phi_{x, u}}{d u}\right|_{u=0} \end{equation}\]

Covolution#

The characteristic function is a great tool for working with linear combination of random variables.

  • if \(x\) and \(y\) are independent random variables then the characteristic function of the linear combination \(a x + b y\) (\(a\) and \(b\) are constants) is

(12)#\[\begin{equation} \Phi_{ax+bx,u} = \Phi_{x,a u}\Phi_{y,b u} \end{equation}\]

  • which means, if \(x\) and \(y\) are independent, the characteristic function of \(x+y\) is the product

(13)#\[\begin{equation} \Phi_{x+x,u} = \Phi_{x,u}\Phi_{y,u} \end{equation}\]

  • The characteristic function of \(ax+b\) is

(14)#\[\begin{equation} \Phi_{ax+b,u} = e^{iub}\Phi_{x,au} \end{equation}\]

Inversion#

There is a one-to-one correspondence between cumulative distribution functions and characteristic functions, so it is possible to find one of these functions if we know the other.

Continuous distributions#

The inversion formula for these distributions is given by

(15)#\[\begin{equation} {\mathbb P}\left(x\right) = \frac{1}{2\pi}\int_{-\infty}^\infty e^{-iuk}\Phi_{x, u} du \end{equation}\]

Discrete distributions#

In these distributions, the random variable \(x\) takes integer values. For example, the Poisson distribution is discrete. The inversion formula for these distributions is given by

(16)#\[\begin{equation} {\mathbb P}\left(x=k\right) = \frac{1}{2\pi}\int_{-\pi}^\pi e^{-iuk}\Phi_{x, u} du \end{equation}\]
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