Math 6040 • Probability

Let \(X_{n}, Y_{n}\) be positive, in \(L^{1}\), and measurable with respect to the filtration \(\mathcal{F}_{n}\). Suppose

\[
\mathbb{E}[X_{n+1} \mid \mathcal{F}_{n}] \le (1 + Y_{n})X_{n}
\]

with \(\sum Y_{n} < \infty\) a.s.
\begin{enumerate}

\item
Show that \(\prod_{i=1}^{n}(1 + Y_{i})\) converges a.s. to a finite limit.



\item
Show

\[
M_{n} =
\frac{X_{n}}{\prod_{i=0}^{n-1}(1 + Y_{i})}
\]
is a super-martingale.


\item
Use (a) and (b) to prove that \(X_{n}\) converges a.s. to a finite limit.

\end{enumerate}