Proof of the product formula for the moment-generating function of a sum of independent random variables

Proof

We write $e^{tS}$ as a product and argue by induction on the number of factors. Truncation and monotone convergence handle infinite values.

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 $W_j = e^{tX_j}$, a random variable with values in $(0, \infty)$. By the exponential of a sum, $e^{tS} = W_1 W_2 \cdots W_n$.
3. By the definition of the expected value of a nonnegative variable, $M_{X_j}(t) = E[W_j] \in (0, \infty]$ and $M_S(t) = E[W_1 \cdots W_n]$.
4. 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 $\mathcal{P}_m$.
5. 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.
6. 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 $P(A \cap C) = P(X_m \in B) \prod_{j < m} P(X_j \in B_j)$.
7. 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 $A \in \mathcal{P}_m$.
8. Let $\mathcal{D}$ be the family of events $A$ with $P(A \cap C) = P(A) P(C)$. It contains $\Omega$.
9. 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 $A' \setminus A \in \mathcal{D}$.
10. 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, $P(A \cap C) = \lim_k P(A_k) P(C) = P(A) P(C)$.
11. By steps 8 to 10, $\mathcal{D}$ is a Dynkin system that contains $\mathcal{P}_m$. By Dynkin's π-λ theorem, $\mathcal{G}_m \subseteq \mathcal{D}$.
12. Let $V_m = W_1 \cdots W_{m-1}$. For $j < m$, the function $x \mapsto e^{tx}$ is continuous, so the preimage of a Borel set under it is Borel. Each event $\set{W_j \in D}$ with $D$ Borel has the form $\set{X_j \in B_j}$ and lies in $\mathcal{G}_m$.
13. So each $W_j$ with $j < m$ is measurable with respect to $\mathcal{G}_m$. A product of measurable functions is measurable, so $V_m$ is $\mathcal{G}_m$-measurable.
14. For an integer $N \ge 1$, let $V_{m,N} = \min(V_m, N)$ and $W_{m,N} = \min(W_m, N)$. Each event $\set{V_{m,N} \in D}$ with $D$ Borel lies in $\mathcal{G}_m$.
15. Each event $\set{W_{m,N} \in D'}$ with $D'$ Borel has the form $\set{X_m \in B}$ for a Borel set $B$. By step 11, $V_{m,N}$ and $W_{m,N}$ are independent.
16. Both are bounded, so they are integrable. By the product rule for independent variables, $E[V_{m,N} W_{m,N}] = E[V_{m,N}] \, E[W_{m,N}]$.
17. As $N \to \infty$, $V_{m,N}$ increases to $V_m$, $W_{m,N}$ increases to $W_m$, and $V_{m,N} W_{m,N}$ increases to $V_m W_m$. All are nonnegative.
18. By the monotone convergence theorem, the three expectations in step 16 converge in $[0, \infty]$ to $E[V_m W_m]$, $E[V_m]$ and $E[W_m]$.
19. The sequences $E[V_{m,N}]$ and $E[W_{m,N}]$ are nondecreasing and positive. So their product tends to $E[V_m] \, E[W_m]$ in $(0, \infty]$. Hence $E[V_m W_m] = E[V_m] \, E[W_m]$.
20. Since $V_m W_m = V_{m+1}$ with $V_{n+1} = W_1 \cdots W_n$, step 19 gives $E[V_{m+1}] = E[V_m] \, E[W_m]$ for $2 \le m \le n$.
21. By induction on $m$, starting from $V_2 = W_1$, we get $E[V_{m+1}] = E[W_1] \cdots E[W_m]$ for $1 \le m \le n$.
22. For $m = n$, steps 2, 3 and 21 give $M_S(t) = M_{X_1}(t) \, M_{X_2}(t) \cdots M_{X_n}(t)$ in $(0, \infty]$.