Math 6010 • Linear Models

Suppose \((M_n)\) is a martingale with \(M_0=0\) and
\(|M_{n+1}-M_n|\leq1\) for all \(n\geq0\).
\begin{enumerate}
\item[(a)] Prove that there exists a finite positive constant \(C\) such that
\(e^x\leq1+x+Cx^2\) for all \(x\in[-1,1]\).
\item[(b)] Prove that \(\E[e^{\lambda M_n}]\leq(1+C\lambda^2)^n\leq e^{C\lambda^2n}\)
for all \(n\geq1\) and \(\lambda\in(0,1)\).
\item[(c)] Prove that \(M_n/n^\delta\to0\) almost surely for every \(\delta>1/2\).
\end{enumerate}