Math 6040 • Probability

Let \(Y_{1}, Y_{2}, \ldots\) be nonnegative i.i.d. random variables with \(\mathbb{E}[Y_{m}] = 1\) and \(P(Y_{m} = 1) <\)
1.

\begin{enumerate}

\item
Show that \(X_{n} = \prod_{m \le n}Y_{m}\) defines a martingale.


\item
Show that \(X_{n} \to 0\) a.s. (Hint: note that \(|X_{n+1} - X_{n}| = X_{n}|Y_{n+1} - 1|\).)


\item
Use the strong law of large numbers to conclude \(\frac{1}{n}\log(X_{n}) \to c < 0\).

\end{enumerate}