Characteristic Function¶

The library makes heavy use of the 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 \({\mathbb P}_x\), where \({\mathbb P}_x\) is the distribution measure of \(x\)

\[\begin{equation} \Phi_{x,u} = {\mathbb E}\left[e^{i u x}\right] = \int e^{i u s} {\mathbb P}_x\left(ds\right) \end{equation}\]

If the distribution of \(x\) has a probability density \(f_x\), then the characteristic function is the Fourier transform of \(f_x\):

\[\begin{equation} \Phi_{x,u} = \int e^{i u s} f_x\left(s\right) ds \end{equation}\]

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

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

Convolution¶

The characteristic function is a great tool for working with linear combinations 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

\[\begin{equation} \Phi_{ax+by,u} = \Phi_{x,a u}\Phi_{y,b u} \end{equation}\]

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

\[\begin{equation} \Phi_{x+y,u} = \Phi_{x,u}\Phi_{y,u} \end{equation}\]

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

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

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

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

Discrete distributions¶

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

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

Characteristic Exponent¶

The characteristic exponent \(\phi_{x,u}\) is defined as

\[\begin{equation} \Phi_{x,u} = e^{-\phi_{x,u}} \end{equation}\]