Math 6040 • Probability

Answer the following questions.
\begin{enumerate}

\item
State the definition of convergence in probability. Using only this definition, show
that \(X_{n} \to X\) in probability if and only if \(\mathbb{E}[1 \wedge |X_{n} - X|] \to 0\). Do not appeal to any
theorems that trivialize the problem.


\item
State the definition of almost sure convergence. Prove that if \(X_{n} \to X\) almost
surely, then \(X_{n} \to X\) in probability. Again, do not appeal to any theorems that
trivialize the problem.

\end{enumerate}