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
Suppose that \(f_n(x) \to f(x)\) and \(f_n \geq 0\), \(\int f_n d\mu = 1\) for all \(n\) and \(f \geq 0\), \(\int f d\mu = 1\). Use Jensen’s inequality and the dominated convergence theorem to show that for any measurable set \(A\), \(\int_A f_n d\mu \to \int_A f d\mu\).
The fact that \[ |f(x) - f_n(x)| = f_n(x) - f(x) + 2max\{0, f(x) - f_n(x)\} \] might be useful.
Problem 4
Show that if for some \(u > 0\), \(E[|X|^u ] < \infty\) and \(E[|Y|^u ] < \infty\), then for any \(q \in (0, u/2)\), \[ \lim_{a \to \infty} a^{q} P \left(|XY| > a\right) = 0 \]
Use Markov’s inequality and the Cauchy-Schwarz inequality.