Math 6040 • Probability

Let \(X_{1}, X_{2}, \ldots\) be i.i.d. with characteristic function \(\phi\).
\begin{enumerate}

\item
Show that if \(\phi'(0) = ia\) and \(S_{n} = X_{1} + \cdots + X_{n}\), then \(S_{n}/n \to a\) in probability.
(Hint: first show that if \(c_{n} \to c \in \mathbb{C}\), then \((1 + c_{n}/n)^{n} \to e^{c}\).)


\item
Show that if \(S_{n}/n \to a\) in probability, then \(\phi(t/n)^{n} \to e^{iat}\) as \(n \to \infty\) through the
integers.

\end{enumerate}