ECON 626: Problem Set 1
Problem 1
Let \(\Omega = \{a,b,c,d\}\).
- Is \(\{\{a\}, \{c\}, \{a,b\}, \emptyset\}\) a \(\sigma\)-field?
- What is the smallest \(\sigma\)-field containing \(\{\{a,b\},\{a,c\}\}\)?
Problem 2
Song (2021) exercise 4.1.
For a collection \(\mathscr{C}\) of sets, we write \(\sigma (\mathscr{C})\) to denote the smallest \(\sigma\)-field that contains \(\mathscr{C}\), and say that \(\sigma (\mathscr{C})\) is the \(\sigma\)-field generated by \(\mathscr{C}\).
Let \(X\) be a random variable on \((\Omega ,\mathscr{F})\) and let \(\mathscr{G}\) be the collection of the sets of the form \(\{\omega \in \Omega :X(\omega )\in B\}\) with \(B\in \mathscr{B}(\mathbf{R})\). Then show that \(\mathscr{G}\) is a \(\sigma\)-field.
Show that \(\{X^{-1}(A):A\in \sigma (\mathscr{C})\}=\sigma(\{X^{-1}(A):A\in \mathscr{C}\})\) for any subset \(\mathscr{C}\) of \(\mathscr{B}(\mathbf{R})\).
Problem 3
Show that if for some \(u ≥ 2\), \(E[|X|^u ] < \infty\) and \(E[|Y|^u ] < \infty\), then for some \(s \in (0, 1)\), \[ \lim_{a \to \infty} a^{1−s} P \left(|XY| > a\right) = 0 \]
Use Markov’s inequality and the Cauchy-Schwarz inequality.
I won’t change the question now (it comes from a problem set for this course from a few years ago, before I was teaching it), but a perhaps better statement of the problem would be to show that \[ \lim_{a \to \infty} a^{q} P \left(|XY| > a\right) = 0 \] for \(q<u/2\). Also, the requirement that \(u \geq 2\) could be replaced with just \(u>0\). Of course, this result implies the result the question askes for.