Prove Theorem 2.34. That is, suppose that (X_{1}, ldots, X_{n}) be a sequence of independent random variables

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Prove Theorem 2.34. That is, suppose that \(X_{1}, \ldots, X_{n}\) be a sequence of independent random variables where \(X_{i}\) has cumulant generating function \(c_{i}(t)\) for \(i=1, \ldots, n\). Then prove that the cumulant generating function of

\[S_{n}=\sum_{i=1}^{n} X_{i}\]

is

\[c_{S_{n}}(t)=\sum_{i=1}^{n} c_{i}(t)\]

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