Math 6040 • Probability

Let \(\{X_{n}\}_{n \ge 1}\) be i.i.d. Normal random variables with mean 0 and variance 1. Let \(S_{0} = 0\)
and \(S_{n} = X_{1} + \cdots + X_{n}\) for \(n \ge 1\). Fix \(t \in \mathbb{R}\) and for \(n \ge 0\) let
\[
M_{n} = e^{tS_{n} - t^{2}n/2}.
\]
\begin{enumerate}

\item
Prove that \(M_{n}\) is a mean one martingale relative to the filtration \(\mathcal{F}_{n} = \sigma(X_{1}, \ldots, X_{n})\).


\item
Prove that if \(\max_{j \le n} S_{j} \ge nt\), then \(\max_{j \le n} M_{j} \ge e^{t^{2}n/2}\).


\item
Conclude that \(P\{\max_{j \le n} S_{j} \ge nt\} \le e^{-t^{2}n/2}\) and then that for any \(c > \theta > 1\) we
have with probability one: \(\exists k_{0} \ge 1\) such that for \(k \ge k_{0}\)
\[
\frac{\max_{j \le \theta^{k}} S_{j}}
{\sqrt{2c\theta^{k-1}\log\log \theta^{k-1}}}
\le 1.
\]



\item
Deduce that almost surely
\[
\limsup_{n \to \infty} \frac{S_{n}}{\sqrt{2n\log\log n}} \le 1.
\]


\end{enumerate}