How to show that for a random variable $X$, $E\left(\sup_{\theta \ \in \ [-B,B]} |\log \Phi(x\theta)|\right) < \infty$ for $\Phi \sim N(0,1)$?

Question

Suppose that $X$ is a random variable with support on $\mathbb{R}$ and that it is from a distribution $p(dx)$. Now assume that we have a parameter $\theta \in [-B,B]$ for some $B > 0$. Assuming that $\Phi(x)$ is the CDF of a standard normal distribution, I would like to show that:

$$ \int \sup_{\theta \ \in \ [-B,B]} |\log \Phi(x\theta)|p(dx) < \infty $$

and so is

$$ \int \sup_{\theta \ \in \ [-B,B]} |\log \Phi(-x\theta)|p(dx) < \infty $$

In other words, the expectation of the supremum of $|\log \Phi(x\theta)|$ and $|\log \Phi(-x\theta)|$ on $\theta \ \in \ [-B,B]$ is bounded.

Does anyone see an easy way to do this? Thanks.

Answer 1

since $\Phi$ and $\log$ are increasing functions such that $\Phi(x) < 1$ for all $x \in \Bbb R$ and $\log(x)<0$ for $x <1$ we have that $$ \sup_{\theta \ \in \ [-B,B]} |\log \Phi(x\theta)| = - \log \Phi(-Bx). $$ Now note that $\lim_{x \to \infty} -\log \Phi(-Bx) = \infty$, so whether or not the integrals are finite depend on the properties of the distribution $p$. In particular, a heavy right tail of $p$ could cause the first integral to be infinite.

Similar considerations can be made for the second integral.