91Ó°ÊÓ

Show that \(L(t)=\log t\) is slowly varying but \(t^{\epsilon}\) is not if \(\epsilon \neq 0\)

Let \(X_{1}, X_{2}, \ldots\) be i.i.d. with \(E X_{i}=0,0<\operatorname{var}\left(X_{i}\right)<\infty\), and let \(S_{n}=\) \(X_{1}+\cdots+X_{n}\). (a) Use the central limit theorem and Kolmogorov's zero- one law to conclude that limsup \(S_{n} / \sqrt{n}=\infty\) a.s. (b) Use an argument by contradiction to show that \(S_{n} / \sqrt{n}\) does not converge in probability. Hint: Consider \(n=m !\).

Show that the distribution of a bounded r.v. \(Z\) is infinitely divisible if and only if \(Z\) is constant. Hint: Show \(\operatorname{var}(Z)=0\).

Show that if \(X_{1}, \ldots, X_{n}\) are independent and uniformly distributed on \((-1,1)\), then for \(n \geq 2, X_{1}+\cdots+X_{n}\) has density $$ f(x)=\frac{1}{\pi} \int_{0}^{\infty}(\sin t / t)^{n} \cos t x d t $$ Although it is not obvious from the formula, \(f\) is a polynomial in each interval \((k, k+1), k \in \mathbf{Z}\) and vanishes on \([-n, n]^{c}\).

(i) Show that if \(X\) is symmetric stable with index \(\alpha\) and \(Y \geq 0\) is an independent stable with index \(\beta<1\) then \(X Y^{1 / \alpha}\) is symmetric stable with index \(\alpha \beta\). (ii) Let \(W_{1}\) and \(W_{2}\) be independent standard normals. Check that \(1 / W_{2}^{2}\) has the density given in (7.14) and use this to conclude that \(W_{1} / W_{2}\) has a Cauchy distribution.

The Ky Fan metric on random variables is defined by $$ \alpha(X, Y)=\inf \\{\epsilon \geq 0: P(|X-Y|>\epsilon) \leq \epsilon\\} $$ Show that if \(\alpha(X, Y)=\alpha\) then the corresponding distributions have Lévy distance \(\rho(F, G) \leq \alpha\).

If \(X_{1}, X_{2}, \ldots\) are independent and have characteristic function \(\exp \left(-|t|^{\alpha}\right)\) then \(\left(X_{1}+\cdots+X_{n}\right) / n^{1 / \alpha}\) has the same distribution as \(X_{1}\).

Find independent r.v.'s \(X, Y\), and \(Z\) so that \(Y\) and \(Z\) do not have the same distribution but \(X+Y\) and \(X+Z\) do.