Prove Theorem 2.33. That is, suppose that \(X_{1}, \ldots, X_{n}\) be a sequence of independent random variables where \(X_{i}\) has characteristic function \(\psi_{i}(t)\), for \(i=1, \ldots, n\). Prove that the characteristic function of

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

is

\[\psi_{S_{n}}(t)=\prod_{i=1}^{n} \psi_{i}(t)\]

Further, prove that if \(X_{1}, \ldots, X_{n}\) are identically distributed with characteristic function \(\psi(t)\) then the characteristic function of \(S_{n}\) is \(\psi_{S_{n}}(t)=\psi^{n}(t)\).

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Theorem 2.33. Let X1,..., Xn be a sequence of independent random vari- ables where X, has characteristic function (t), for i = 1,...,n, then the characteristic function of n Sn = Xi, Xi, i=1 is n vs,(t) = [[vi(t). i=1 If X1,..., Xn are identically distributed with characteristic function (t) then the characteristic function of Sn is s(t) = " (t).