Proof of the product formula for the characteristic function of a sum of independent random variables

Proof

We write $e^{itS}$ as a product and argue by induction on the number of factors. We split each complex factor into its real and imaginary parts.

1. Let $X_1, \dots, X_n$ be independent real random variables on $(\Omega, \mathcal{F}, P)$, and let $S = X_1 + \dots + X_n$. Fix $t \in \R$.
2. For each $j$, let $U_j = e^{itX_j}$. By Euler's formula, $U_j = \cos(tX_j) + i \sin(tX_j)$, so its real and imaginary parts are bounded by $1$.
3. For a complex variable $Z = A + iB$ with bounded real variables $A$ and $B$, write $E[Z] = E[A] + i E[B]$. By the definition of the characteristic function, $\varphi_{X_j}(t) = E[U_j]$ and $\varphi_S(t) = E[e^{itS}]$.
4. By the product of exponentials, $e^{itS} = U_1 U_2 \cdots U_n$.
5. For $2 \le m \le n$, let $\mathcal{P}_m$ be the family of events $\bigcap_{j < m} \set{X_j \in B_j}$ with Borel sets $B_j \subseteq \R$. Let $\mathcal{G}_m$ be the σ-algebra generated by $\mathcal{P}_m$.
6. The intersection of two such events is $\bigcap_{j < m} \set{X_j \in B_j \cap B_j'}$, again in $\mathcal{P}_m$. So $\mathcal{P}_m$ is a π-system.
7. Fix a Borel set $B$ and let $C = \set{X_m \in B}$. For $A = \bigcap_{j < m} \set{X_j \in B_j}$ in $\mathcal{P}_m$, the definition of independence with $J = \set{1, \dots, m}$ gives $P(A \cap C) = P(X_m \in B) \prod_{j < m} P(X_j \in B_j)$.
8. The same definition with $J = \set{1, \dots, m - 1}$ gives $\prod_{j < m} P(X_j \in B_j) = P(A)$. So $P(A \cap C) = P(A) P(C)$ for each $A \in \mathcal{P}_m$.
9. Let $\mathcal{D}$ be the family of events $A$ with $P(A \cap C) = P(A) P(C)$. It contains $\Omega$.
10. If $A \subseteq A'$ are in $\mathcal{D}$, then $P((A' \setminus A) \cap C) = P(A' \cap C) - P(A \cap C) = (P(A') - P(A)) P(C)$. This is $P(A' \setminus A) P(C)$, so $A' \setminus A \in \mathcal{D}$.
11. If $A_1 \subseteq A_2 \subseteq \cdots$ are in $\mathcal{D}$ with union $A$, then $A_k \cap C$ increases to $A \cap C$. By continuity of probability, $P(A \cap C) = \lim_k P(A_k) P(C) = P(A) P(C)$.
12. By steps 9 to 11, $\mathcal{D}$ is a Dynkin system that contains $\mathcal{P}_m$. By Dynkin's π-λ theorem, $\mathcal{G}_m \subseteq \mathcal{D}$.
13. Claim: if $Y$ is a bounded real $\mathcal{G}_m$-measurable variable and $g$ is a bounded continuous real function, then $E[Y g(X_m)] = E[Y] \, E[g(X_m)]$.
14. Proof of the claim. Each event $\set{Y \in D}$ with $D$ Borel lies in $\mathcal{G}_m$. Each event $\set{g(X_m) \in D'}$ equals $\set{X_m \in g^{-1}(D')}$, and $g^{-1}(D')$ is Borel.
15. By step 12, $Y$ and $g(X_m)$ are independent. Both are bounded, hence integrable. The product rule for independent variables gives the claim.
16. Let $V_m = U_1 \cdots U_{m-1}$, and write $V_m = A_m + i B_m$ with $A_m$, $B_m$ real. Since $\abs{U_j} = 1$, we have $\abs{V_m} = 1$, so $A_m$ and $B_m$ are bounded by $1$.
17. Expanding the product, $A_m$ and $B_m$ are sums of products of the variables $\cos(tX_j)$ and $\sin(tX_j)$ with $j < m$. Fix $j < m$ and a Borel set $D$. The set $B_j = \set{x \in \R : \cos(tx) \in D}$ is Borel, since $x \mapsto \cos(tx)$ is continuous.
18. So the event $\set{\cos(tX_j) \in D} = \set{X_j \in B_j}$ lies in $\mathcal{P}_m$, with all other sets equal to $\R$. The same holds for $\sin$, so these variables are $\mathcal{G}_m$-measurable.
19. Sums and products of measurable functions are measurable. So $A_m$ and $B_m$ are $\mathcal{G}_m$-measurable.
20. Let $c = \cos(tX_m)$ and $s = \sin(tX_m)$. Then $V_m U_m = (A_m c - B_m s) + i (A_m s + B_m c)$.
21. By linearity of expectation and the claim of step 13, $E[V_m U_m] = (E[A_m] E[c] - E[B_m] E[s]) + i (E[A_m] E[s] + E[B_m] E[c])$.
22. The right side equals $(E[A_m] + i E[B_m]) (E[c] + i E[s]) = E[V_m] \, E[U_m]$.
23. Since $V_m U_m = V_{m+1}$, with $V_{n+1} = U_1 \cdots U_n$, step 22 gives $E[V_{m+1}] = E[V_m] \, E[U_m]$ for $2 \le m \le n$.
24. By induction on $m$, starting from $V_2 = U_1$, we get $E[V_{m+1}] = E[U_1] \cdots E[U_m]$ for $1 \le m \le n$.
25. For $m = n$, steps 3, 4 and 24 give $\varphi_S(t) = \varphi_{X_1}(t) \, \varphi_{X_2}(t) \cdots \varphi_{X_n}(t)$.