Characteristic Function

The characteristic function of a Random Variable $X$ is:

$${\varphi }_X(t)=\mathbb{E}[e^{itX}]=\mathbb{E}[\cos (tX)]+i\mathbb{E}[\sin (tX)],$$

When $X$ has Density $f$,

$${\varphi }_X(t)={\int }_{\mathbb{R}}{e}^{ixt}f(x)\mathrm{d}x,$$

which is similar to the Fourier transform of a function, except for the absence of a minus sign in the exponent.

The characteristic function has a similar form to the MGF. However, it addresses the boundedness issue of MGF, e.g., for Cauchy Distribution, as it always holds that $|{\varphi }_X(t)|\le 1$.

Due to this similarity, CF has the same properties as MGF, such as the Inversion Theorem and the following:

• $Y=aX+b⟹M_Y(s)=e^{sb}{M}_X(as)$
• $X\perp Y⟹M_{X+Y}(s)=M_X(s)M_Y(s)$
• $X\perp Y⟹M_{X,Y}(s,t)=M_X(s)M_Y(t)$
• $X\perp Y\wedge (Z=BX+(1-B)Y,B\sim \mathrm{Bern}(p))⟹M_X(s)=p{M}_X(s)+(1-p)M_Y(s)$

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Other Properties

Inversion Theorem for Continuous Random Variables

For a univariate continuous random variable $X$ with CF ${\varphi }_X(t)$, we have explicit inversion formula:

$$f_X(x)=\frac{1}{2\pi }\underset{T\to \infty }{\lim }{\int }_{-T}^T{e}^{-itx}{\varphi }_X(t)\mathrm{d}t.$$

Again, this is similar to the inversion formula for the Fourier transform.

DCT Applies

Since the CF is always bounded by 1, the Dominated Convergence Theorem applies, which gives

$$\underset{n\to \infty }{\lim }{\varphi }_{X_n}={\varphi }_X,$$

given that $X_n\to X$ a.s.

Generating Moments

Similar to MGF, we can generate moments from CF:

$$\mathbb{E}[X^k]=i^{-k}\frac{{\mathrm{d}}^k}{\mathrm{d}{t}^k}{\varphi }_X(t){|}_{t=0}.$$