91Ó°ÊÓ

(i) Show that if \(X\) and \(Y\) are independent then \(\sigma(X)\) and \(\sigma(Y)\) are. (ii) Conversely, if \(\mathcal{F}\) and \(\mathcal{G}\) are independent, \(X \in \mathcal{F}\), and \(Y \in \mathcal{G}\), then \(X\) and \(Y\) are independent.

Let \(P\) be a probability measure on \((\Omega, \mathcal{F})\) (i) monotonicity. If \(A \subset B\) then \(P(B)-P(A)=P(B-A) \geq 0\). (ii) subadditivity. If \(A_{m} \in \mathcal{F}\) for \(m \geq 1\) and \(A \subset \cup_{m=1}^{\infty} A_{m}\) then \(P(A) \leq\) \(\sum_{m=1}^{\infty} P\left(A_{m}\right)\) (iii) continuity from below. If \(A_{i} \uparrow A\) (i.e., \(A_{1} \subset A_{2} \subset \ldots\) and \(\cup_{i} A_{i}=A\) ) then \(P\left(A_{i}\right) \uparrow P(A)\). (iv) continuity from above. If \(A_{i} \downarrow A\) (i.e., \(A_{1} \supset A_{2} \supset \ldots\) and \(\cap_{i} A_{i}=A\) ) then \(P\left(A_{i}\right) \downarrow P(A)\).

The \(L^{2}\) weak law generalizes immediately to certain dependent sequences. Suppose \(E X_{n}=0\) and \(E X_{n} X_{m} \leq r(n-m)\) for \(m \leq n\) (no absolute value on the left-hand side!) with \(r(k) \rightarrow 0\) as \(k \rightarrow \infty\). Show that \(\left(X_{1}+\ldots+X_{n}\right) / n \rightarrow 0\) in probability.

Let \(X_{0}=(1,0)\) and define \(X_{n} \in \mathbf{R}^{2}\) inductively by declaring that \(X_{n+1}\) is chosen at random from the ball of radius \(\left|X_{n}\right|\) centered at the origin, i.e., \(X_{n+1} /\left|X_{n}\right|\) is uniformly distributed on the ball of radius 1 and independent of \(X_{1}, \ldots, X_{n} .\) Prove that \(n^{-1} \log \left|X_{n}\right| \rightarrow c\) a.s. and compute \(c .\)

Show (a) that \(d(X, Y)=E(|X-Y| /(1+|X-Y|))\) defines a metric on the set of random variables, i.e., (i) \(d(X, Y)=0\) if and only if \(X=Y\) a.s., (ii) \(d(X, Y)=d(Y, X)\), (iii) \(d(X, Z) \leq d(X, Y)+d(Y, Z)\) and (b) that \(d\left(X_{n}, X\right) \rightarrow 0\) as \(n \rightarrow \infty\) if and only if \(X_{n} \rightarrow X\) in probability.

A function \(f\) is said to be lower semicontinuous or l.s.c. if $$ \liminf _{y \rightarrow x} f(y) \geq f(x) $$ and upper semicontinuous (u.s.c.) if \(-f\) is l.s.c. Show that \(f\) is l.s.c. if and only if \(\\{x: f(x) \leq a\\}\) is closed for each \(a \in \mathbf{R}\) and conclude that semicontinuous functions are measurable.

One-sided Chebyshev bound. (i) Let \(a>b>0,00\). Show that \(P(Y \geq a) \leq\) \(\sigma^{2} /\left(a^{2}+\sigma^{2}\right)\), and there is a \(Y\) for which equality holds.

Prove that \(P\left(\lim \sup A_{n}\right) \geq \limsup P\left(A_{n}\right)\) and $$ P\left(\liminf A_{n}\right) \leq \liminf P\left(A_{n}\right) $$ For independent events, however, the necessary condition for \(P\left(\lim\right.\) sup \(\left.A_{n}\right)>0\) is sufficient for \(P\left(\lim \sup A_{n}\right)=1\).

A function \(\varphi: \Omega \rightarrow \mathbf{R}\) is said to be simple if $$ \varphi(\omega)=\sum_{m=1}^{n} c_{m} 1_{A_{m}}(\omega) $$ where the \(c_{m}\) are real numbers and \(A_{m} \in \mathcal{F}\). Show that the class of \(\mathcal{F}\) measurable functions is the smallest class containing the simple functions and closed under pointwise limits.

Use the fact that a gamma \((1, \lambda)\) is an exponential with parameter \(\lambda\), and induction to show that the sum of \(n\) independent exponential \((\lambda)\) r.v.'s, \(X_{1}+\cdots+X_{n}\), has a gamma \((n, \lambda)\) distribution.