Math 6010 • Linear Models

Let \(Z_n\) be a martingale difference with respect to a filtration
\(\mathcal{F}_n\), i.e. \(\E[Z_{n+1}\mid\mathcal{F}_n]=0\) for all \(n\geq0\).
Assume the following hold:
\begin{enumerate}
\item[1)] \(n^{-1/2}\max_{m\leq n}|Z_m|\to0\) in probability.
\item[2)] \(n^{-1}\sum_{m=1}^{n}|Z_m|^2\to1\) in probability.
\item[3)] \(\sum_{m=1}^{n}|Z_m|^2\leq2n\) almost surely, for all \(n\).
\end{enumerate}
Prove that \(M_n=n^{-1/2}\sum_{m=1}^{n}Z_m\) converges in distribution to a
standard normal.

\textbf{Hint:} Prove that \(|1+ix|\leq e^{x^2}\) for all \(x\in\mathbb{R}\) and
\(e^{ix}=(1+ix)e^{-x^2/2+r(x)}\) with \(|r(x)|\leq x^3\) for all
\(x\in\mathbb{R}\). Let
\[
  T_n=\prod_{m=1}^{n}\left(1+itn^{-1/2}Z_m\right)
\]
and
\[
  U_n=
  e^{-\frac{t^2}{2n}\sum_{m=1}^{n}Z_m^2
  +\sum_{m=1}^{n}r(tn^{-1/2}Z_m)}.
\]
Compute \(\E[T_n]\) then prove that \(\E[T_nU_n]\to e^{-t^2/2}\) and conclude.