Characteristic Function

Convolutional Integral

To compute the distribution for $Z = X + Y$, we have: $$ f_Z(z) = \int_{\mathbb{R}} f_X(x) f_Y(z - x) dx $$ The new pdf is the convolutional integral of $X, Y$. However, for $X_1, \dots, X_n$ it might be hard to compute the integral.

Characteristic Function Basics

Note that similar to CDF, the characteristic function is unique for every random variable $X$.

Properties

The convolutional integral can correspond to operations in terms of characteristic functions. $$ \begin{aligned} f_Z(z) &= \int_{\mathbb{R}} f_X(x)f_Y(z - x)dx \\ \phi_Z(t) &= \mathbb{E}\left[ e^{itZ} \right] \\ &= \int_{\mathbb{R}} f_Z(z)e^{itz} dz \\ &= \int_{\mathbb{R}} e^{itz}\int_{\mathbb{R}} f_X(x)f_Y(z - x)dx dz \\ &= \int_{\mathbb{R}} e^{it(x + y)}\int_{\mathbb{R}} f_X(x)f_Y(y)dx dy \\ &= \int_{\mathbb{R}} f_X(x) \int_{\mathbb{R}} e^{it(x + y)}f_Y(y) dy dx \\ &= \int_{\mathbb{R}} e^{itx}f_X(x) \int_{\mathbb{R}} e^{ity}f_Y(y) dy dx \\ &= \int_{\mathbb{R}} e^{itx}f_X(x) dx \int_{\mathbb{R}} e^{ity}f_Y(y) dy \\ &= \phi_X(t) \cdot \phi_Y(t) \\ \phi_{X + Y} &= \phi_X(t) \cdot \phi_Y(t) = \phi_{X \ast Y} \\ \phi_{X_1 + \cdots + X_n} &= \phi_{X_1}(t) \cdots \phi_{X_n}(t) \end{aligned} $$ Reference:

characteristic function of a convolution of measures

To CDF

Note that there is a one-to-one correspondence between CDF and CF! $$ \begin{aligned} f_X(x) &= \frac{1}{2\pi} \int_{\mathbb{R}} e^{-itx} \phi_X(t) dt \end{aligned} $$ Reference: