Characteristic function of a sum of independent random variables

Statement

Let $X_1, \dots, X_n$ be independent real random variables, and let $S = X_1 + \dots + X_n$. Then for each $t \in \mathbb{R}$, the characteristic functions satisfy $$\varphi_S(t) = \varphi_{X_1}(t) \, \varphi_{X_2}(t) \cdots \varphi_{X_n}(t).$$

Remarks

Proof: $e^{itS} = e^{itX_1} \cdots e^{itX_n}$. The factors are independent and bounded, so the product rule for expectations applies to their real and imaginary parts. For i.i.d. $X_i$, $\varphi_S(t) = \varphi_{X_1}(t)^n$. The sample mean has $\varphi_{S/n}(t) = \varphi_{X_1}(t/n)^n$. The converse is false: $\varphi_{X+Y} = \varphi_X \varphi_Y$ does not imply that $X$ and $Y$ are independent.

Explanations

L1 human

Adding independent variables corresponds to multiplying their characteristic functions. This is the key step in the usual proof of the central limit theorem.