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It is well known that if $X,Y$ are independent random variables on $(\Omega,\mathscr{F},P)$ with respective characteristic functions $\varphi_X,\varphi_Y$, then $\varphi_{X+Y}=\varphi_X\varphi_Y$.

If $X_i$ are independent random variables with characteristics $\varphi_i$, is it true that the characteristic function of $\sum_{i=1}^\infty X_i$ (assuming the sum exists a.s.) is $\prod_{i=1}^\infty \varphi_i$?

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Since $S_n=\sum\limits_{k=1}^nX_k$ converges almost surely to $S=\sum\limits_{k=1}^\infty X_k$, $\varphi_{S_n}\to\varphi_S$ pointwise by dominated convergence. By independence, $\varphi_{S_n}=\prod\limits_{k=1}^n\varphi_k$. Thus, $\prod\limits_{k=1}^n\varphi_k$ converges pointwise when $n\to\infty$ and its limit is both $\prod\limits_{k=1}^\infty\varphi_k$ and $\varphi_S$. This shows that $\varphi_S=\prod\limits_{k=1}^\infty\varphi_k$.

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