Math 6010 • Linear Models

Let \(p_n\) and \(q_n\) denote two probability mass functions on
\(\mathbb{Z}^n\), for every integer \(n\geq1\). That is, for all \(n\geq1\):
\begin{enumerate}
\item[(a)] \(p_n(\mathbf{x})\geq0\) and \(q_n(\mathbf{x})\geq0\) for all
\(\mathbf{x}\in\mathbb{Z}^n\); and
\item[(b)] \(\sum_{\mathbf{x}\in\mathbb{Z}^n}p_n(\mathbf{x})
=\sum_{\mathbf{x}\in\mathbb{Z}^n}q_n(\mathbf{x})=1\).
\end{enumerate}
Now suppose \(\{X_i\}_{i=1}^{\infty}\) is a collection of random variables such
that \((X_1,\ldots,X_n)\) has mass function \(p_n\) for every \(n\geq1\). That is,
\[
  \Prob\{X_1=x_1,\ldots,X_n=x_n\}
  =
  p_n(x_1,\ldots,x_n)
  \qquad
  \text{for all }(x_1,\ldots,x_n)\in\mathbb{Z}^n.
\]
\begin{enumerate}
\item[(a)] Prove that \(p_n(X_1,\ldots,X_n)>0\) a.s. for all \(n\geq1\).
\item[(b)] Prove that \(\{Z_n\}_{n=1}^{\infty}\) is a martingale, where, for all
\(n\geq1\),
\[
  Z_n:=
  \frac{q_n(X_1,\ldots,X_n)}{p_n(X_1,\ldots,X_n)},
\]
almost surely on the \(\Prob\)-measure-one event
\(\{p_n(X_1,\ldots,X_n)>0\}\), and \(Z_n:=0\) otherwise.
\item[(c)] Prove that \(\lim_{n\to\infty}Z_n\) exists a.s. and is finite a.s.
(This is a starting point for likelihood-ratio testing in classical
statistics.)
\end{enumerate}